Last updated at April 19, 2021 by Teachoo

Transcript

Example 34 Find two positive numbers whose sum is 15 and the sum of whose squares is minimum. Let first number be ๐ Since Sum of two positive numbers is 15 ๐ฅ+ 2nd number = 15 2nd number = 15 โ ๐ Let S(๐ฅ) be the sum of the squares of the numbers S(๐ฅ)= (1st number)2 + (2nd number) 2 S(๐)=๐^๐+(๐๐โ๐)^๐ We need to minimize S(๐) Finding Sโ(๐) Sโ(๐ฅ)=๐(๐ฅ^2+ (15 โ ๐ฅ)^2 )/๐๐ฅ =๐(๐ฅ^2 )/๐๐ฅ+(๐(15 โ ๐ฅ)^2)/๐๐ฅ = 2๐ฅ+ 2(15โ๐ฅ)(โ1) = 2๐ฅโ 2(15โ๐ฅ) = 2๐ฅโ30+2๐ฅ = 4๐โ๐๐ Putting Sโ(๐)=๐ 4๐ฅโ30=0 4๐ฅ=30 ๐ฅ=30/4 ๐=๐๐/๐ Finding Sโโ(๐) Sโโ(๐ฅ)=๐(4๐ฅ โ 30)/๐๐ฅ = 4 Since Sโโ(๐)>๐ at ๐ฅ=15/2 โด ๐ฅ=15/2 is local minima Thus, S(๐ฅ) is Minimum at ๐ฅ=15/2 Hence, 1st number = ๐ฅ=๐๐/๐ 2nd number = 15โ๐ฅ=15โ15/2=๐๐/๐

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Chapter 6 Class 12 Application of Derivatives

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.