Find two positive numbers whose sum is 15 and the sum of whose sq… - Vidyadip

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 $S(x) = x^2 + (15 – x)^2 = 2x^2 – 30x + 225$
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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