Question

Find the two positive numbers whose sum is $15$ and sum of whose squares is minimum.

Answer 1

$a+b=15$

$f(a,b)=a^2+b^2$ is minimum

$f\left(a\right)=a^2+{(15-a)}^2$

$f^{\prime }\left(a\right)=2a-2(15-a)=4a-30=0$

$a=\frac{15}{2},b=\frac{15}{2}$

So $a=\frac{15}{2},\frac{15}{2}$ are two numbers for which $f(a,b)=a^2+b^2$ is minimum