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.

Then, we get,

P(x) = x(24 –x) = 24x – x²

⇒ P’(x) = 24 – 2x

⇒ P’’(x) = -2

Now, P’(x) = 0

⇒ x = 12

And

P’’(12) = -2 <0

Then, by second derivative test,

x = 12 is the point of local maxima of P.

Therefore, the product of the numbers is the maximum when the numbers are 12 and 24 – 12 = 12.