Example 22 Find two positive numbers whose sum is 15 and the sum of whose squares is minimum. Let first number be š’™ Since Sum of two positive numbers is 15 š‘„+ 2nd number = 15 2nd number = 15 ā€“ š’™ Let S(š‘„) be the sum of the squares of the numbers S(š‘„)= (1st number)2 + (2nd number) 2 S(š’™)=š’™^šŸ+(šŸšŸ“āˆ’š’™)^šŸ We need to minimize S(š’™) Finding Sā€™(š’™) Sā€™(š‘„)=š‘‘(š‘„^2+ (15 āˆ’ š‘„)^2 )/š‘‘š‘„ =š‘‘(š‘„^2 )/š‘‘š‘„+(š‘‘(15 āˆ’ š‘„)^2)/š‘‘š‘„ = 2š‘„+ 2(15āˆ’š‘„)(āˆ’1) = 2š‘„āˆ’ 2(15āˆ’š‘„) = 2š‘„āˆ’30+2š‘„ = 4š’™āˆ’šŸ‘šŸŽ Putting Sā€™(š’™)=šŸŽ 4š‘„āˆ’30=0 4š‘„=30 š‘„=30/4 š’™=šŸšŸ“/šŸ Finding Sā€™ā€™(š’™) Sā€™ā€™(š‘„)=š‘‘(4š‘„ āˆ’ 30)/š‘‘š‘„ = 4 Since Sā€™ā€™(š’™)>šŸŽ at š‘„=15/2 āˆ“ š‘„=15/2 is local minima Thus, S(š‘„) is Minimum at š‘„=15/2 Hence, 1st number = š‘„=šŸšŸ“/šŸ 2nd number = 15āˆ’š‘„=15āˆ’15/2=šŸšŸ“/šŸ

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.