Question

Find the two numbers whose sum is 12 and whose product is maximum.

Answer 1

If first number is, say $x$, another number will be $12-x$.

The multiplication of two number makes a quadratic function.

$x(12-x)$ is a quadratic function.

$-x^2+12x$

$-(x^2-12x)$

$-(x^2-12x+36-36)$

$-(x-6)^2+36$

Therefore, the maximum product is 36 at x=6.

Thus two number will be equal to 6 and 6.