Getting on For all the preceding functions, discuss, whenever possible, whether local min and max are global.

Size: px
Start display at page:

Download "Getting on For all the preceding functions, discuss, whenever possible, whether local min and max are global."

Transcription

1 Problems Warming up Find the domain of the following functions, establish in which set they are differentiable, then compute critical points. Apply 2nd order sufficient conditions to qualify them as local ma, min, saddle or can t tell.. f, y = 3 + y y. 2. f, y = lny 2y 2 3. f, y = y 2 2y. f, y = y 3 y 2 + 3y + 2y f, y = 3 + y 5 3 3y f, y = 2 8y + e y 7. f, y = ln 2 + y 2 y 8. f, y = e 2 +y y 9. f, y = 2 y 2 0. f, y = lny 2 2 3y 2. f, y = y f, y = 2 y + y f, y = ln y 2 +y 2 Getting on For all the preceding functions, discuss, whenever possible, whether local min and ma are global.

2 Answer to problems Altogether, we answer the warming up and the getting on part. Solution to # The function f, y = 3 + y y, is defined by a polinomial, hence is differentiable infinte times, with continuous derivatives on R 2. Gradient is + y + 3 f, y = 2 + y 2 + 2y Critical points are solutions to + y y 2 = 0 + 2y = 0 that is the set Z = 0,, 0; 0, 23 }, 2 ; 2, 2 Hessian matri is D 2 6 2y + f, y = 2y + 2 det D 2 f0, 0 = 0 = 6 < 0, 0; 0 saddle point. 2 det D 2 f0, 0 = = 6 < 0, 0; saddle point. 3 det D 2 f2, 2 = = 32 > 0 and a, = 8 > 0, 2; 2 local min. 3 det D 2 f 2 3, 2 = = 32 3 > 0 and a, = 8 < 0, 2 3 ; 2 local ma. 3 Local are not global, ma and min. To prove it, it suffices to observe that and inf R f, = f, = while sup f, = + R Solution to #2. The function f, y = lny 2y 2, Hessian is defined on the open set D =, y R 2 : > 0, y > 0 }, y R 2 : < 0, y < 0 } where it is infinite times continuously differentiable. Gradient is f, y = y + y Critical points are solutions to = 0 y + y = 0 2

3 which solutions are [ [ =, y = 2], and =, y = 2] so that the set of critical points is singleton Z =, } 2 Hessian matri is D 2 f, y = y 2 so that D 2 f 0, 2 = which is negative definite,, local ma. Note that no global min eists: if it eisted, as D is an open set, it would be also a local min, but no local min eists. On the other hand, there is no global ma either. Indeed, for < 0 and y = we have lim f, = lim f t, = lim t + ln t + t 2 = +. t + Remark. Note that the function is *not* concave on D, since D is not a conve set; f is concave on the subset of the domain, y R 2 : > 0, y > 0 } where it attains maimum, but the same conclusion cannot be drawn on the whole domain. Solution to #3 f, y = y 2 2y,, 2 2 = 0 Gradient is f, y =, 2y 2 = = 0 Solution is: [ =, y = ] 2y 2 = Hessian is, f conve, global min 0 2 Ma does not eist f, y = 2 + y 2 2 so that Solution to # lim f, = = + sup f = + R 2. f, y = y 3 y 2 + 3y + 2y 3 2, 6 + 3y Gradient is f, y = 3 2y 3y y = 0 [ Critical points 3 2y 3y = 0, = 97+ 2, y = Hessian is D 2 f, y = 3 6y , determinant: 3 97 < 0 saddle, determinant: 3 97 a local ma 3 ] [, = 97 2, y = ] 97 2

4 Not a global ma, for sup f = lim f0, a = lim a 3 a 2 2a = + R 2 a + a + No global min otherwise it should be a critical point. Indeed we can say more, that is inf f = lim f0, a = lim a 3 a 2 + 2a = R 2 a + a + Solution to #5 f, y = 3 + y 5 3 3y + 2 2,, Gradient is f, y = 3 y = 0 Critical points 3 y 5 2, 3 = 0 Solution is: [ = 3, y = 6], [ =, y = 6], [ = 3, y = ], [ =, y = ] 6 0 Hessian is D 2 f, y =, 0 6y D 2 f 3, 6 =, determinant: 2, saddle point D 2 f, 6 =, determinant: 2, local min, not globlal for inf f = D 2 f 3, = D 2 f, = Solution to #6 f, y = 2 8y + e y y 2 8 Gradient is 8 + e y + 2 Critical points, determinant: 2, local ma, not a global ma for sup f = +, determinant: 2, saddle point y 2 8 = e y + 2 = 0, if y = = 0, Solution is: = 5, = 5 + if = e y 6 = 0, y = ln 6 2y 2 8 Hessian is D 2 f = 2 8 e y D 2 f 5, 0 = D 2 f + 5, 0 = 5, 0; + 5, 0,, ln , determinant: 60, saddle point, determinant: 60, saddle point

5 2 ln D 2 f, ln 6 = ln e Solution to #7. f, y = ln 2 + y 2 y lim f a, = lim a +, determinant: 32 ln 6 > 0, local min, but not global, as [ a 2 + 8a + e ] = inf f = a + The domain of f is D = R 2 \ 0, 0}, while the gradient is 2 f, y = 2 + y 2 y, 2y 2 + y 2 Note also that: - partial derivatives are defined and continuous at every point in D, - D is an open set then f is differentiable in D. Critical points are those P, y such that f, y = 0, that is, the set of solutions of the system 2 2 +y 2 y = 0 2y 2 +y 2 = 0 Case : Assume 0 and y 0 we have 2 2 +y y = 0 2 2y 2 +y = y y = y = 2 2y 2y = y 2 2 +y 2 = 2y 2 2 2y 2 = 0 2 +y = 2 2y + y y = 0 2 +y = 2 2y The first equation is satisfied either for a = y or b = y If a pluggin into second equation we get 2y 2 = 2 y2 = y =, y = so that the system is satisfied by, and,. If b, proceding similarly we get 2y 2 = 2 y2 = that has no real solution. Case 2: Let us wonder whether 0, y with y 0, or, 0 with 0 may be a solution. In the first case we get 0 y = 0 2y y 2 0 = 0 that has no solution. Similarly one derives that, 0 with 0 cannot be a solution. Critical points of f are, and,. We now compute Hessian matri. then 2 2 y 2 y D 2 f, y = 2 +y y 2 2 y 2 2 y 2 2 +y y 2 2 D 2 f, = D 2 f, =

6 whose determinant is -, then the matri is indefinte and both critical points are saddle points. Solution to #8 f, y = e 2 +y y. a D = R 2 ; b f is differentiable in D, with gradient f, y = c Critical points are solutions to 2e 2 +y 2 2y = 0 2ye 2 +y 2 2 = 0 2e 2 +y 2 2y, 2ye 2 +y 2 2 The first equation is satisfied if one of the factors is null, that is: = 0, or 2 e 2 +y 2 2y = 0 In case, the 2nd equation gives 2ye y2 = 0 y = 0, that implies 0, 0 is critical. In case 2, we plug 2y = e 2 +y 2 into the 2nd equation, deriving 22y 2 2 = 0 2y 2 2 = 0 2y 2y + = 0, hence either 2a = 2y or 2b = 2y In both cases, plugging = ± 2y into the st equation and assuming, y 0, 0 without loss of generality, as 0,0 has been already computed as a solution, we derive e 3y2 = 2y. Now we prove that such equation has no real solution, by showing the function ϕy := e 3y2 2y, has no zeroes ϕ in R. In particular it suffices to show that ϕ is strictly positive everywhere. Surely ϕ has no zero in, 0] because in such interval e 3y2 > 0 while 2y 0, which implies ϕy > 0 for all y 0. There eist no zero of ϕ also in ]0, 2 ], that is for 0 < y /2, as y > 0 e 3y2 > and 2y so that y > 0 ϕy = e 3y2 2y > = 0, y ]0, 2 ]. What is left to check is the zeroes in ]/2, + [. Note that if we show that ϕ y > 0 for all y > /2 that is, ϕ increasing in ]/2, + [ we may derive that no zero eists in ]/2, + [, as ϕ/2 > 0 and ϕ increasing in ]/2, + [ implies ϕy ϕ/2 > 0, for all y ]/2, + [. Then we show now ϕ y > 0 for all y > /2. Note that ϕ y = 6ye 32 2, so that ϕ /2 = 3e 3/ 2 > 0. Moreover ϕ y = e 3y y 2 > 0 for all y. Then ϕ is itself increasing and positive in /2, the it is strictly positive for all y > /2. The only critical point is 0, 0. d Hessian is D 2 f, y = 2e 2 +y y ye 2 +y 2 ye 2 +y 2 2e 2 +y 2 + 2y 2 D 2 f0, 0 = Since f 0, 0 = 2 > 0, e det D 2 f0, 0 = > 0 we derive 0, 0 is a local min. Global minimization of f Weierstrass Theorem is needed, see lecture # We now discuss whether 0;0 is also a GLOBAL min. The idea is to prove that the originale problem is equivalent to maimizing f on a compact region. 6

7 As eponential function e 2 +y 2 grows, positively, at infinity faster than 2 y one may guess that indeed a global min does eist and is attained in a neighborhood of the origin. Nevertheless the assertion has to be precisely proven. We make use here of definition of global min and of Weierstrass Theorem in R n. If we show that f is coercive, that is in this case lim f = + + we may then infer that the f attains its minimum on R 2, and that the local minimizer is indeed a global minimizer. And consequently 0; 0 is shown to be the unique global minimum point for f on R 2. Why? If f coercive, there eists a radius R 0 > 0 such that inf R 2 f = inf B R0 f = min B R0 f where B R0 =, y R 2 : 2 + y 2 R 2 0}. Hence inf is indeed a min by means of Weierstrass Theorem, because f is continuous and B R0 is a compact subset of R n. Moreover, that is also a global min for f on the entire R 2. Ultimately, the minimizer has to be a critical point of f also, as it is a minimum attained on an open unbounded region. We then show that f is coercive. Let R > 0, and assume = R. Then 2 y R 3 so that for all, y B R 0; 0 one has f, y e R2 2R 3 =: ϕr Passing to limits as R +, we derive lim f, y lim + which implies that f is coercive. Solution to #9 ϕr = R + lim e R2 2R 3 = + R + f, y = 2 y 2 The domain is D =, y : y + y 0} =, y : y > 0 and + y > 0}, y : y 0 and + y 0} but f is differentiable only at D =, y : y + y > 0} =, y : y > 0 and + y > 0}, y : y < 0 and + y < 0} At any couple of type, or,, f attains the value 0 which is the minimum attainable value, hence global min points are, : R}, : R} Such points may not be met among critical points, as the function f is not differentiable at those points. A global ma does not eist, as lim f, 0 = lim =

8 Gradient is Critical points 2 y 2 y 2 y 2 Solution to #0. f, y = lny 2 2 3y 2, Domain D =, y : y > 2 + } y 2 Gradient is 2 6y + y 2 2 y 2 Critical points 2 = 0 6y + y 2 2 = 0, Solution is: P 0, P 2 0, / D, D Hessian is D 2 f 0, y 2 2 y y y = , whose determinant is: = , Then P 2 is a local ma. P 2 is the unique global maimizer. Alternative very epensive proof : f, y < ln y 3y 2 ma y> ln y 3y2 = f P 2 We now show that P 2 is a global ma. Indeed, let 0, y 0, y : y = }. Then lim f, y =,y ln = 0,y 0 so that there eists ε > 0 for instance, one may choose ε = 0.0 such that where the closed set D ε is given by for in the complement set sup D f = sup f D ε D ε =, y D : y ε}, y D D ε =, y D : < y < ε} f, y < ln ε 3y 2 < ln ε = ln 0.0 =. 60 < fp

9 Let s now show that If we do so then lim f, y =,y +,,y D sup D f = sup f = sup f D ε D ε B where B is a suitable closed ball centered at the origin and, being D ε B closed and bounded, hence compact, f would attain its ma in D ε B, hence in D, hence at P 2. Now note that y > 0, y + + y + + y + so that, if y +, then f, y < ln y 3y 2 ; if instead +, then y +, same conclusion, so that is proved. Solution to # Trivial: without any computation, 3, is the global minimizer and no local ma eists; sup f = +. Solution to #2 f, y = 2 y + y + 2,, The function is defined on D =, y : y 0} but differentiable in D o =, y : y > 0} 2 Gradient is y 2 y 2 y + 2 Critical points y 2 y = 0 y + = 0, Solution is: [ = 2, y = ] 8 Hessian is 2 = [ y2 y y 2 y 2 2 y 3 2 [ 3 ] ] = 2,y= 8 whose determinant is 2, so that 2, 8 is a local min. Nonetheless, it is not a global min. Indeed f 2, 8 = [ 2 ] y + y + 2 = = 2,y= 6 8 [ f, 00 = 2 ] y + y + 2 = 39 = Solution to #3 f, y = ln a Domain: y 2 +y 2. D =, y R 2 : =,y= 00 y 2 + y 2 > 0, 2 + y 2 0} =, y R 2 : y > 0,, y 0, 0} =, y R 2 : > 0, y > 0}, y R 2 : < 0, y < 0} 9

10 b f, y = y y 2, 2 y 2 2 +y 2 y differentiable in all D D open set and partial derivatives are continuous at every point of D. c The set of critical points Z is made of the solutions to that is y y 2 = 0 y 2 2 = 0 2 y 2 2 +y 2 y = 0 2 y 2 = 0 Z =, y D : y y + = 0} y y + = 0 =, y D : y = }, y D : y = } =, y D : y = } =, y D : y = } Let us try to apply 2nd order cpnditions. The Hessian matri is D 2 f, y = 2 y 2 y 2 +y y 2 +y 2 2 y 2 +y 2 2 y 2 2 +y y 2 y + and computed at points of Z becomes D 2 f, = so that f, < 0, but det Hf, = 0, so that 2nd order sufficient conditions do not apply. From 2nd order necessary conditions, we know that critical points cannot be local min, but could be ma or saddle. Note further that the Hessian is negative or null in the domain det D 2 f, y = y 2 + y 2 y y so that it is indefinite not conve nor concave in D Z, and negative semidefinite in Z. Indeed all points in Z are GLOBAL ma. Note that, since on D we have y > 0, then 0 < y 2 + y 2 2 as the second inequality is true y 2 0. Moreover 2 is attained at all points of Z. Hence f attains its GLOBAL ma with ma f = ln 2. The property is very easy to guess, nonetheless one may not get it at first sight. The one may start analyzing level sets Γ k of f, discovering that they are of type 2 + y 2 2αy = 0, with α =. Then, 2e k by a change of coordinates, one checks that the curve 2 + y 2 2αy = 0 describes two lines through the origin, so that all level curves are indeed of type, β for some positive β. Since f, β = ln β + β 2 and such function of β attains its ma at on the positive real ais, then the maimizing level curve was ;. 0

Lecture 5 Principal Minors and the Hessian

Lecture 5 Principal Minors and the Hessian Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

More information

Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x

Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written

More information

36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous? 36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

More information

MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2

MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2 MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

Practice with Proofs

Practice with Proofs Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

More information

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all. 1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

More information

DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents

DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition

More information

About the Gamma Function

About the Gamma Function About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is

More information

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)! Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

More information

t := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).

t := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d). 1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction

More information

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

More information

Special Situations in the Simplex Algorithm

Special Situations in the Simplex Algorithm Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

Lectures 5-6: Taylor Series

Lectures 5-6: Taylor Series Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

More information

1 Maximizing pro ts when marginal costs are increasing

1 Maximizing pro ts when marginal costs are increasing BEE12 Basic Mathematical Economics Week 1, Lecture Tuesda 12.1. Pro t maimization 1 Maimizing pro ts when marginal costs are increasing We consider in this section a rm in a perfectl competitive market

More information

Calculus 1st Semester Final Review

Calculus 1st Semester Final Review Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim

More information

BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBERT SPACE REVIEW BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

More information

Adaptive Online Gradient Descent

Adaptive Online Gradient Descent Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

More information

Limits and Continuity

Limits and Continuity Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

More information

Homework # 3 Solutions

Homework # 3 Solutions Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8

More information

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper. FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

More information

Duality of linear conic problems

Duality of linear conic problems Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

More information

A FIRST COURSE IN OPTIMIZATION THEORY

A FIRST COURSE IN OPTIMIZATION THEORY A FIRST COURSE IN OPTIMIZATION THEORY RANGARAJAN K. SUNDARAM New York University CAMBRIDGE UNIVERSITY PRESS Contents Preface Acknowledgements page xiii xvii 1 Mathematical Preliminaries 1 1.1 Notation

More information

Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

More information

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

More information

LIMITS AND CONTINUITY

LIMITS AND CONTINUITY LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

More information

Convex analysis and profit/cost/support functions

Convex analysis and profit/cost/support functions CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m

More information

THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE. Contents THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

More information

3. Reaction Diffusion Equations Consider the following ODE model for population growth

3. Reaction Diffusion Equations Consider the following ODE model for population growth 3. Reaction Diffusion Equations Consider the following ODE model for population growth u t a u t u t, u 0 u 0 where u t denotes the population size at time t, and a u plays the role of the population dependent

More information

Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

Metric Spaces Joseph Muscat 2003 (Last revised May 2009) 1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

More information

0 0 such that f x L whenever x a

0 0 such that f x L whenever x a Epsilon-Delta Definition of the Limit Few statements in elementary mathematics appear as cryptic as the one defining the limit of a function f() at the point = a, 0 0 such that f L whenever a Translation:

More information

5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1

5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1 MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing

More information

Some stability results of parameter identification in a jump diffusion model

Some stability results of parameter identification in a jump diffusion model Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss

More information

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price

More information

4 Lyapunov Stability Theory

4 Lyapunov Stability Theory 4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We

More information

PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.

PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm. PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle

More information

3. Mathematical Induction

3. Mathematical Induction 3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

More information

Metric Spaces. Chapter 7. 7.1. Metrics

Metric Spaces. Chapter 7. 7.1. Metrics Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

More information

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a 88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

More information

AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:

AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period: AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be

More information

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

More information

Date: April 12, 2001. Contents

Date: April 12, 2001. Contents 2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........

More information

Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

More information

Scalar Valued Functions of Several Variables; the Gradient Vector

Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,

More information

Roots of equation fx are the values of x which satisfy the above expression. Also referred to as the zeros of an equation

Roots of equation fx are the values of x which satisfy the above expression. Also referred to as the zeros of an equation LECTURE 20 SOLVING FOR ROOTS OF NONLINEAR EQUATIONS Consider the equation f = 0 Roots of equation f are the values of which satisfy the above epression. Also referred to as the zeros of an equation f()

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

More information

Notes on metric spaces

Notes on metric spaces Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

More information

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part

More information

Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.

Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3. 5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2

More information

The Mean Value Theorem

The Mean Value Theorem The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers

More information

Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.

Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}. Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian

More information

Mathematical Induction

Mathematical Induction Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

More information

Solutions of Equations in One Variable. Fixed-Point Iteration II

Solutions of Equations in One Variable. Fixed-Point Iteration II Solutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

More information

sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2

sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2 . Problem Show that using an ɛ δ proof. sin() lim = 0 Solution: One can see that the following inequalities are true for values close to zero, both positive and negative. This in turn implies that On the

More information

Rolle s Theorem. q( x) = 1

Rolle s Theorem. q( x) = 1 Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question

More information

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

More information

THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION

THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION ERICA CHAN DECEMBER 2, 2006 Abstract. The function sin is very important in mathematics and has many applications. In addition to its series epansion, it

More information

Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let H and J be as in the above lemma. The result of the lemma shows that the integral Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

More information

Differentiating under an integral sign

Differentiating under an integral sign CALIFORNIA INSTITUTE OF TECHNOLOGY Ma 2b KC Border Introduction to Probability and Statistics February 213 Differentiating under an integral sign In the derivation of Maximum Likelihood Estimators, or

More information

Math 2443, Section 16.3

Math 2443, Section 16.3 Math 44, Section 6. Review These notes will supplement not replace) the lectures based on Section 6. Section 6. i) ouble integrals over general regions: We defined double integrals over rectangles in the

More information

CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

More information

The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method

The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method Robert M. Freund February, 004 004 Massachusetts Institute of Technology. 1 1 The Algorithm The problem

More information

160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

More information

Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

More information

Shape Optimization Problems over Classes of Convex Domains

Shape Optimization Problems over Classes of Convex Domains Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY e-mail: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore

More information

Section 1.3 P 1 = 1 2. = 1 4 2 8. P n = 1 P 3 = Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3,..., = 1 2 4.

Section 1.3 P 1 = 1 2. = 1 4 2 8. P n = 1 P 3 = Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3,..., = 1 2 4. Difference Equations to Differential Equations Section. The Sum of a Sequence This section considers the problem of adding together the terms of a sequence. Of course, this is a problem only if more than

More information

Representation of functions as power series

Representation of functions as power series Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

More information

What is Linear Programming?

What is Linear Programming? Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to

More information

0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup

0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup 456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),

More information

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

More information

2008 AP Calculus AB Multiple Choice Exam

2008 AP Calculus AB Multiple Choice Exam 008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

More information

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

More information

Finite dimensional topological vector spaces

Finite dimensional topological vector spaces Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.

Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima. Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

Lecture 7: Finding Lyapunov Functions 1

Lecture 7: Finding Lyapunov Functions 1 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1

More information

Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Exam #3 Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

More information

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

More information

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

More information

Exercises in Mathematical Analysis I

Exercises in Mathematical Analysis I Università di Tor Vergata Dipartimento di Ingegneria Civile ed Ingegneria Informatica Eercises in Mathematical Analysis I Alberto Berretti, Fabio Ciolli Fundamentals Polynomial inequalities Solve the

More information

MINIMIZATION OF ENTROPY FUNCTIONALS UNDER MOMENT CONSTRAINTS. denote the family of probability density functions g on X satisfying

MINIMIZATION OF ENTROPY FUNCTIONALS UNDER MOMENT CONSTRAINTS. denote the family of probability density functions g on X satisfying MINIMIZATION OF ENTROPY FUNCTIONALS UNDER MOMENT CONSTRAINTS I. Csiszár (Budapest) Given a σ-finite measure space (X, X, µ) and a d-tuple ϕ = (ϕ 1,..., ϕ d ) of measurable functions on X, for a = (a 1,...,

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

More information

Implicit Differentiation

Implicit Differentiation Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some

More information

Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

More information

Taylor and Maclaurin Series

Taylor and Maclaurin Series Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

More information

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

More information

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the

More information

5.1 Derivatives and Graphs

5.1 Derivatives and Graphs 5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has

More information

1 Norms and Vector Spaces

1 Norms and Vector Spaces 008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

More information

So let us begin our quest to find the holy grail of real analysis.

So let us begin our quest to find the holy grail of real analysis. 1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

More information