Find two positive numbers whose sum is 14 and the sum of whose sq… - Vidyadip

Question

Find two positive numbers whose sum is $14$ and the sum of whose squares is minimum.

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Answer

Let the numbers be $x$ and $y.$ Then,
$x + y = 14...(i)$
Let S be the sum of the squares of $x$ and $y.$ Then,
$S=x^2+y^2$
$\Rightarrow S = x ^2+(14- x )^2$
$\Rightarrow S=2 x ^2-28 x +196$
$\Rightarrow \frac{d S}{d x}=4 x -28$ and $\frac{d^2 S}{d x^2}=4$
The critical points of $S$ are given by $\frac{d S}{d x}=0$.
$\because \frac{d S}{d x}=0$
$\Rightarrow 4 x -28=0$
$\Rightarrow x =7$
Clearly $\frac{d^2 S}{d x^2}=4 ;0$
Thus, $S$ is minimum when $x = 7$ Putting $x = 7$ in equation $(i),$
we obtain $y = 7$
Hence. the required numbers are both equal to $7.$

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