Example 34 - Find two numbers whose sum is 15, sum of squares

Example 34 - Chapter 6 Class 12 Application of Derivatives - Part 2
Example 34 - Chapter 6 Class 12 Application of Derivatives - Part 3
Example 34 - Chapter 6 Class 12 Application of Derivatives - Part 4

  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

Example 34 Find two positive numbers whose sum is 15 and the sum of whose squares is minimum. Let first number be ๐’™ Since Sum of two positive numbers is 15 ๐‘ฅ+ 2nd number = 15 2nd number = 15 โ€“ ๐’™ Let S(๐‘ฅ) be the sum of the squares of the numbers S(๐‘ฅ)= (1st number)2 + (2nd number) 2 S(๐’™)=๐’™^๐Ÿ+(๐Ÿ๐Ÿ“โˆ’๐’™)^๐Ÿ We need to minimize S(๐’™) Finding Sโ€™(๐’™) Sโ€™(๐‘ฅ)=๐‘‘(๐‘ฅ^2+ (15 โˆ’ ๐‘ฅ)^2 )/๐‘‘๐‘ฅ =๐‘‘(๐‘ฅ^2 )/๐‘‘๐‘ฅ+(๐‘‘(15 โˆ’ ๐‘ฅ)^2)/๐‘‘๐‘ฅ = 2๐‘ฅ+ 2(15โˆ’๐‘ฅ)(โˆ’1) = 2๐‘ฅโˆ’ 2(15โˆ’๐‘ฅ) = 2๐‘ฅโˆ’30+2๐‘ฅ = 4๐’™โˆ’๐Ÿ‘๐ŸŽ Putting Sโ€™(๐’™)=๐ŸŽ 4๐‘ฅโˆ’30=0 4๐‘ฅ=30 ๐‘ฅ=30/4 ๐’™=๐Ÿ๐Ÿ“/๐Ÿ Finding Sโ€™โ€™(๐’™) Sโ€™โ€™(๐‘ฅ)=๐‘‘(4๐‘ฅ โˆ’ 30)/๐‘‘๐‘ฅ = 4 Since Sโ€™โ€™(๐’™)>๐ŸŽ at ๐‘ฅ=15/2 โˆด ๐‘ฅ=15/2 is local minima Thus, S(๐‘ฅ) is Minimum at ๐‘ฅ=15/2 Hence, 1st number = ๐‘ฅ=๐Ÿ๐Ÿ“/๐Ÿ 2nd number = 15โˆ’๐‘ฅ=15โˆ’15/2=๐Ÿ๐Ÿ“/๐Ÿ

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.