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Ex 6.5, 16 - Find two positive numbers whose sum is 16 - Minima/ maxima (statement questions) - Number questions

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  1. Chapter 6 Class 12 Application of Derivatives
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Ex 6.5,16 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 = ļ·ļ·š‘“š‘–š‘Ÿš‘ š‘” š‘›š‘¢š‘šš‘š‘’š‘Ÿ ļ·Æļ·®3ļ·Æ+ļ·ļ·š‘ š‘’š‘š‘œš‘›š‘‘ š‘›š‘¢š‘šš‘š‘’š‘Ÿ ļ·Æļ·®3ļ·Æ Let Sļ·š‘„ļ·Æ = š‘„3 + ļ·ļ·16āˆ’š‘„ļ·Æļ·®3ļ·Æ We Need to Find Minimum Value of sļ·š‘„ļ·Æ Finding Sā€™ļ·š‘„ļ·Æ Sā€™ļ·š‘„ļ·Æ= ļ·š‘‘ļ·ļ·š‘„ļ·®3ļ·Æ+ ļ·ļ·16 āˆ’ š‘„ļ·Æļ·®3ļ·Æļ·Æļ·®š‘‘š‘„ļ·Æ = 3š‘„2 + 3ļ·ļ·16āˆ’š‘„ļ·Æļ·®2ļ·Æ. ļ·0āˆ’1ļ·Æ = 3š‘„2 + 3ļ·ļ·16āˆ’š‘„ļ·Æļ·®2ļ·Æļ·āˆ’1ļ·Æ = 3š‘„2 ā€“ 3ļ·ļ·ļ·16ļ·Æļ·®2ļ·Æ+ļ·ļ·š‘„ļ·Æļ·®2ļ·Æāˆ’2ļ·16ļ·Æļ·š‘„ļ·Æļ·Æ = 3š‘„2 ā€“ 3ļ·256+ļ·š‘„ļ·®2ļ·Æāˆ’32š‘„ļ·Æ = 3š‘„2 ā€“ 3ļ·256ļ·Æāˆ’3ļ·š‘„ļ·®2ļ·Æ+3ļ·32ļ·Æš‘„ = ā€“3ļ·256āˆ’32š‘„ļ·Æ Putting Sā€™ļ·š‘„ļ·Æ=0 ā€“3ļ·256āˆ’32š‘„ļ·Æ=0 256 ā€“ 32š‘„ = 0 32š‘„ = 256 š‘„ = ļ·256ļ·®32ļ·Æ š‘„ = 8 Finding Sā€™ā€™ļ·š‘„ļ·Æ Sā€™ļ·š‘„ļ·Æ=āˆ’3ļ·256āˆ’32š‘„ļ·Æ Sā€™ā€™ļ·š‘„ļ·Æ=ļ·š‘‘ļ·āˆ’3ļ·256 āˆ’ 32š‘„ļ·Æļ·Æļ·®š‘‘š‘„ļ·Æ = ā€“3 ļ·š‘‘ļ·256 āˆ’ 32š‘„ļ·Æļ·®š‘‘š‘„ļ·Æ = ā€“3 ļ·0āˆ’32ļ·Æ = 96 > 0 Since Sā€™ā€™ļ·š‘„ļ·Æ>0 for š‘„ = 8 ā‡’ š‘„ = 8 is point of local minima & Sļ·š‘„ļ·Æ is minimum at š‘„ = 8 Hence, 1st number = x = 8 & 2nd number = 16 ā€“ x = 16 ā€“ 8 = 8

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