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

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:

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

$\therefore P^{\prime}(x)=24-2 x$

$P^{\prime \prime}(x)=-2$

$P^{n}(x)=-2$

Now,

$P^{\prime}(x)=0 \Rightarrow x=12$

Also,

$P^{\prime \prime}(12)=-2<0$

$\therefore$ 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 $12=12$.