Question

Find two numbers whose sum is $24$ and whose product is as large as possible.

Answer 1

Let one number be x. Then, the other number is $(24 − x)$. Let $P(x)$ denote the product of the two numbers. Thus, we have:

$p(x) =x(24-x)24x-x^{2}$

$p'(x)=24-2x$

$p''(x)=-2$

Now,

$p'(x)=0⇒x=12$

Also,

$p''(12)=-2<0$

∴By second derivative test, $x=12$ is the point of local maxima of P. Hence, the product of the numbers is the maximum when the numbers are 12 and $24−12=12.$