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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
Question
Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.
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Answer

Let one of the numbers be x. Then the other number is (15 – x).
Let S(x) denote the sum of the squares of these numbers. Then
S(x) = x2 + (15 – x)2 = 2x2 – 30x + 225
or $\left\{\begin{array}{l} {S^{\prime}(x)=4 x-30} \\ {S^{\prime \prime}(x)=4} \end{array}\right.$
Now S'(x) = 0, gives, x = $\frac{15}{2}$.
Also S''$\left(\frac{15}{2}\right)$ = 4 > 0. 
Therefore, by second derivative test, x = $\frac{15}{2}$ is the point of local minima of S. Hence the sum of squares of numbers is minimum when the numbers are $\frac{15}{2}$ and 15 -$\frac{15}{2}=\frac{15}{2}$.

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