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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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Ex 6.3, 16 Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.Let first number be š‘„ Now, First number + second number =16 š‘„ + second number = 16 second number = 16 ā€“ š‘„ Now, Sum of Cubes = (š‘“š‘–š‘Ÿš‘ š‘” š‘›š‘¢š‘šš‘š‘’š‘Ÿ )^3+(š‘ š‘’š‘š‘œš‘›š‘‘ š‘›š‘¢š‘šš‘š‘’š‘Ÿ )^3 Let S(š‘„) = š‘„3 + (16āˆ’š‘„)^3 We Need to Find Minimum Value of s(š‘„) Finding Sā€™(š‘„) Sā€™(š‘„)= š‘‘(š‘„^3+ (16 āˆ’ š‘„)^3 )/š‘‘š‘„ = 3š‘„2 + 3(16āˆ’š‘„)^2. (0āˆ’1) = 3š‘„2 + 3(16āˆ’š‘„)^2 (āˆ’1) = 3š‘„2 ā€“ 3((16)^2+(š‘„)^2āˆ’2(16)(š‘„)) = 3š‘„2 ā€“ 3(256+š‘„^2āˆ’32š‘„) = 3š‘„2 ā€“ 3(256)āˆ’3š‘„^2+3(32)š‘„ = ā€“3(256āˆ’32š‘„) Putting Sā€™(š‘„)=0 ā€“3(256āˆ’32š‘„)=0 256 ā€“ 32š‘„ = 0 32š‘„ = 256 š‘„ = 256/32 š‘„ = 8 Finding Sā€™ā€™(š‘„) Sā€™(š‘„)=āˆ’3(256āˆ’32š‘„) Sā€™ā€™(š‘„)=š‘‘(āˆ’3(256 āˆ’ 32š‘„))/š‘‘š‘„ = ā€“3 š‘‘(256 āˆ’ 32š‘„)/š‘‘š‘„ = ā€“3 [0āˆ’32] = 96 > 0 Since Sā€™ā€™(š‘„)>0 for š‘„ = 8 š‘„ = 8 is point of local minima & S(š‘„) is minimum at š‘„ = 8 Hence, 1st number = x = 8 & 2nd number = 16 ā€“ x = 16 ā€“ 8 = 8

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.