Using Cauchy Inequality to Find Function Extremums


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1 International Journal of Trend in Scientific Research and Development (IJTSRD) Special Issue on International Research Development and Scientific Excellence in Academic Life Available Online: eissn: Using Cauchy Inequality to Find Function Extremums G. A. Akhmedova, O. Yu. Makhmudova Kokand State Pedagogical Institute, Kokand, Uzbekistan ABSTRACT The article explains the relationships between the mean values using triangles. Here are some ways to determine the extreme values of some functions using the Cauchy inequality, which represents the relationship between arithmetic mean and geometric mean. Consider sections with lengths and. Their average arithmetic is, their average geometry is, and their average harmonic is. We can compare these values in the following ways (Figure 1). KEYWORDS: function extremums, averages, Cauchy inequality 1rasm 2rasm 3rasm h and! =! # = It can be seen from Figure 2 that it is. (The arithmetic mean of and is equal to the mean geometry at =.) From the similarity of triangles & and '!( =! ; &=!!) + or &= ; & < because, (fig. 3). The three mean values are equal to =. Now let's talk about the average values application in solving problems to find the largest values of the function. 1example. At what value of / does the (/)=/(1 /) (0</<1) function take the largest value? Solution: According to inequalities 3example. At what value of / does the function h(/)=4/+ B,/ > 0 take the smallest value? 6 Solution. According to + 2 4/+ B 2DB 6 6 4/=12 4/+ B 6 =12 /=F Answer: min h(/)=h5 F 9=12 4example. At what value of / does (/)= function / > 0 take the largest value? Solution: :6 G H :6 G H= 7 I /=F D 7 :6 6 B = 7 I :6 G H, the /(1 /) = 7 : /(1 /)= 7 : /=7 Answer: max (/)=5 7 9=7 : 2example. At what value of / does the function (/)=/(/ 1) (0</<1) take the smallest value? Solution. If < < = h < > =. /(/ 1)= /(1 /) = 7 : /(/ 1)= 7 : /= 7 Answer: min (/)=5 7 9= 7 : Answer: max (/)=5 F 9=7 I 5example. At what values of (/)= 6, :6JF, the 6 function /> 2 takes the smallest value. Solution: According to + 2 / +4/+53 /+2 6, :6JF 6 =/ /+2 2N(/+2) 49 /+2 =14 =14 /=5 Answer: min (/)=(5)=14 6example. Prove that a rectangle with equal sides has a square with the smallest perimeter. ID: IJTSRD38751 Special Issue on International Research Development and Scientific Excellence in Academic LifePage 114
2 International Journal of Trend in Scientific Research and Development eissn: Let one side of the rectangle be the one length, then the other side of the given surface O= (>0) is, 6. Proof: Let one side of the rectangle be the one length, then the other side of the given surface O= (>0) is, 6. This is the rectangle perimeter P(/)=2(/+, 6 ) It s clear, 25/+, D/, 6 =4. 25/+, 6 9=4 / 2/+ =0 /= min P(/)=4 if 6, = /=. In that case, the rectangles with equal sides 6 have the smallest perimeter. 2 x example. If A =, B = 2, then A > B. 2 x x Proof. A = = x > x + 2 x + 2 So A > B. Proved. References [1] Sivashinsky I.Kh. Inequalities in tasks. M: Science,1967 [2] Zaitsev V. V., Ryzhkov V. V., Skanavi M. I. Elementary mathematics. M: Science,1976 ID: IJTSRD38751 Special Issue on International Research Development and Scientific Excellence in Academic LifePage 115
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