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
  2. Serial order wise

Transcript

Ex 6.5,13 Find two numbers whose sum is 24 and whose product is as large as possible. Let first number be ๐‘ฅ Now, given that First number + Second number = 24 ๐‘ฅ + second number = 24 Second number = 24 โ€“ ๐‘ฅ Product = ๏ท๐‘“๐‘–๐‘Ÿ๐‘ ๐‘ก ๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๏ทฏ ร— ๏ท๐‘ ๐‘’๐‘๐‘œ๐‘›๐‘‘ ๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ๏ทฏ = ๐‘ฅ ๏ท24โˆ’๐‘ฅ๏ทฏ Let P๏ท๐‘ฅ๏ทฏ = ๐‘ฅ ๏ท24โˆ’๐‘ฅ๏ทฏ We need product as large as possible Hence we need to find maximum value of P๏ท๐‘ฅ๏ทฏ Finding Pโ€™(x) P๏ท๐‘ฅ๏ทฏ=๐‘ฅ๏ท24โˆ’๐‘ฅ๏ทฏ P๏ท๐‘ฅ๏ทฏ=24๐‘ฅโˆ’๏ท๐‘ฅ๏ทฎ2๏ทฏ Pโ€™๏ท๐‘ฅ๏ทฏ=24โˆ’2๐‘ฅ Pโ€™๏ท๐‘ฅ๏ทฏ=2๏ท12โˆ’๐‘ฅ๏ทฏ Putting Pโ€™๏ท๐‘ฅ๏ทฏ=0 2๏ท12โˆ’๐‘ฅ๏ทฏ=0 12 โ€“ ๐‘ฅ = 0 ๐‘ฅ = 12 Finding Pโ€™โ€™๏ท๐‘ฅ๏ทฏ Pโ€™๏ท๐‘ฅ๏ทฏ=24โˆ’2๐‘ฅ Pโ€™โ€™๏ท๐‘ฅ๏ทฏ = 0 โ€“ 2 = โ€“ 2 Thus, pโ€™โ€™๏ท๐‘ฅ๏ทฏ < 0 for ๐‘ฅ = 12 โ‡’ ๐‘ฅ = 12 is point of maxima & P๏ท๐‘ฅ๏ทฏ is maximum at ๐‘ฅ = 12 Finding maximum P(x) P๏ท๐‘ฅ๏ทฏ=๐‘ฅ๏ท24โˆ’๐‘ฅ๏ทฏ Putting ๐‘ฅ = 12 p๏ท12๏ทฏ= 12๏ท24โˆ’12๏ทฏ = 12๏ท12๏ทฏ = 144 โˆด First number = x = 12 & Second number = 24 โ€“ x = (24 โ€“ 12)= 12

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.