Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.

Solution:

Let the two positive numbers be $x$ and y. Then,

$x+y=15$    ....(1)

Now,

$z=x^{2}+y^{2}$

$\Rightarrow z=x^{2}+(15-x)^{2}$      [From eq. (1)]

$\Rightarrow z=x^{2}+x^{2}+225-30 x$

$\Rightarrow z=2 x^{2}+225-30 x$

$\Rightarrow \frac{d z}{d x}=4 x-30$

For maximum or minimum values of $\mathrm{z}$, we must have

$\frac{d z}{d x}=0$

$\Rightarrow 4 x-30=0$

$\Rightarrow x=\frac{15}{2}$

$\frac{d^{2} z}{d x^{2}}=4>0$

Substituting $x=\frac{15}{2}$ in $(1)$, we get

$y=\frac{15}{2}$

Thus, $z$ is minimum when $x=y=\frac{15}{2}$.