Question

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

Answer 1

Let the numbers be x and y. Then,

$x+y=14$

Let S be the sum of the squares of x and y. Then,

$S=x^2+y^2$

$S=x^2+(14-x{)}^2$

$S=2x^2-28x+196$

$\frac{dS}{dx}=4x-28$

$\frac{d^{2}S}{d{x}^2}=4$

The critical points of S are given by $\frac{dS}{dx}=0$.

$4x-28=0⟹x=7$

As, $\frac{d^{2}S}{d{x}^2}>0$

Thus, S is minimum when $x=7$. Putting $x=7$, we get, $y=7$.

Hence, the required numbers are both equal to 7.