Last updated at April 19, 2021 by
This video is only available for Teachoo black users Solve all your doubts with Teachoo Black (new monthly pack available now!)
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 |