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 \(24−12=12.\)