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Example 34 - Find two numbers whose sum is 15, sum of squares - Minima/ maxima (statement questions) - Number questions

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Example 34 Find two positive numbers whose sum is 15 and the sum of whose squares is minimum. Let first number be 𝑥 Given sum of two positive number is 15 1st number + 2nd number = 15 𝑥+ 2nd number = 15 2nd number = 15 – 𝑥 Let S﷐𝑥﷯ be the sum of the squares of the numbers ⇒ S﷐𝑥﷯= (1st number)2 + (2nd number) 2 S﷐𝑥﷯=﷐𝑥﷮2﷯+﷐﷐15−𝑥﷯﷮2﷯ We need to minimize S﷐𝑥﷯ Finding S’﷐𝑥﷯ S﷐𝑥﷯=﷐𝑥﷮2﷯+﷐﷐15−𝑥﷯﷮2﷯ S’﷐𝑥﷯=﷐𝑑﷐﷐𝑥﷮2﷯+ ﷐﷐15 − 𝑥﷯﷮2﷯﷯﷮𝑑𝑥﷯ =﷐𝑑﷐﷐𝑥﷮2﷯﷯﷮𝑑𝑥﷯+﷐𝑑﷐﷐15 − 𝑥﷯﷮2﷯﷮𝑑𝑥﷯ = 2𝑥+ 2﷐15−𝑥﷯﷐−1﷯ = 2𝑥− 2﷐15−𝑥﷯ = 2𝑥−30+2𝑥 = 4𝑥−30 Putting S’﷐𝑥﷯=0 4𝑥−30=0 4𝑥=30 𝑥=﷐30﷮4﷯ 𝑥=﷐15﷮2﷯ Finding S’’﷐𝑥﷯ S’﷐𝑥﷯=4𝑥−30 S’’﷐𝑥﷯=﷐𝑑﷐4𝑥−30﷯﷮𝑑𝑥﷯ = 4 Putting 𝑥=﷐15﷮2﷯ in S’’﷐𝑥﷯ S’’﷐﷐15﷮2﷯﷯=4 at 𝑥=﷐15﷮2﷯ ⇒ s’’ ﷐𝑥﷯>0 at 𝑥=﷐15﷮2﷯ ∴ 𝑥=﷐15﷮2﷯ is local minima Thus, S﷐𝑥﷯ is Minimum at 𝑥=﷐15﷮2﷯ Hence, 1st number = 𝑥=﷐𝟏𝟓﷮𝟐﷯ 2nd number = 15−𝑥=15−﷐15﷮2﷯=﷐𝟏𝟓﷮𝟐﷯

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