Ex 6.5, 13 - Find two numbers whose sum is 24, product is large - Minima/ maxima (statement questions) - Number questions

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  1. Chapter 6 Class 12 Application of Derivatives
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Ex 6.5,13 Find two numbers whose sum is 24 and whose product is as large as possible. Let first number be š‘„ Now, given that First number + Second number = 24 š‘„ + second number = 24 Second number = 24 ā€“ š‘„ Product = ļ·š‘“š‘–š‘Ÿš‘ š‘” š‘›š‘¢š‘šš‘š‘’š‘Ÿ ļ·Æ Ɨ ļ·š‘ š‘’š‘š‘œš‘›š‘‘ š‘›š‘¢š‘šš‘š‘’š‘Ÿļ·Æ = š‘„ ļ·24āˆ’š‘„ļ·Æ Let Pļ·š‘„ļ·Æ = š‘„ ļ·24āˆ’š‘„ļ·Æ We need product as large as possible Hence we need to find maximum value of Pļ·š‘„ļ·Æ Finding Pā€™(x) Pļ·š‘„ļ·Æ=š‘„ļ·24āˆ’š‘„ļ·Æ Pļ·š‘„ļ·Æ=24š‘„āˆ’ļ·š‘„ļ·®2ļ·Æ Pā€™ļ·š‘„ļ·Æ=24āˆ’2š‘„ Pā€™ļ·š‘„ļ·Æ=2ļ·12āˆ’š‘„ļ·Æ Putting Pā€™ļ·š‘„ļ·Æ=0 2ļ·12āˆ’š‘„ļ·Æ=0 12 ā€“ š‘„ = 0 š‘„ = 12 Finding Pā€™ā€™ļ·š‘„ļ·Æ Pā€™ļ·š‘„ļ·Æ=24āˆ’2š‘„ Pā€™ā€™ļ·š‘„ļ·Æ = 0 ā€“ 2 = ā€“ 2 Thus, pā€™ā€™ļ·š‘„ļ·Æ < 0 for š‘„ = 12 ā‡’ š‘„ = 12 is point of maxima & Pļ·š‘„ļ·Æ is maximum at š‘„ = 12 Finding maximum P(x) Pļ·š‘„ļ·Æ=š‘„ļ·24āˆ’š‘„ļ·Æ Putting š‘„ = 12 pļ·12ļ·Æ= 12ļ·24āˆ’12ļ·Æ = 12ļ·12ļ·Æ = 144 āˆ“ First number = x = 12 & Second number = 24 ā€“ x = (24 ā€“ 12)= 12

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