Check sibling questions

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


Transcript

Example 34 Find two positive numbers whose sum is 15 and the sum of whose squares is minimum. Let first number be 𝒙 Since Sum of two positive numbers is 15 π‘₯+ 2nd number = 15 2nd number = 15 – 𝒙 Let S(π‘₯) be the sum of the squares of the numbers S(π‘₯)= (1st number)2 + (2nd number) 2 S(𝒙)=𝒙^𝟐+(πŸπŸ“βˆ’π’™)^𝟐 We need to minimize S(𝒙) Finding S’(𝒙) S’(π‘₯)=𝑑(π‘₯^2+ (15 βˆ’ π‘₯)^2 )/𝑑π‘₯ =𝑑(π‘₯^2 )/𝑑π‘₯+(𝑑(15 βˆ’ π‘₯)^2)/𝑑π‘₯ = 2π‘₯+ 2(15βˆ’π‘₯)(βˆ’1) = 2π‘₯βˆ’ 2(15βˆ’π‘₯) = 2π‘₯βˆ’30+2π‘₯ = 4π’™βˆ’πŸ‘πŸŽ Putting S’(𝒙)=𝟎 4π‘₯βˆ’30=0 4π‘₯=30 π‘₯=30/4 𝒙=πŸπŸ“/𝟐 Finding S’’(𝒙) S’’(π‘₯)=𝑑(4π‘₯ βˆ’ 30)/𝑑π‘₯ = 4 Since S’’(𝒙)>𝟎 at π‘₯=15/2 ∴ π‘₯=15/2 is local minima Thus, S(π‘₯) is Minimum at π‘₯=15/2 Hence, 1st number = π‘₯=πŸπŸ“/𝟐 2nd number = 15βˆ’π‘₯=15βˆ’15/2=πŸπŸ“/𝟐

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.