Question

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

Answer 1

Let's assume that one number is $x$

then other number is $24-x$.

Now we have to maximise their product.

So product is given by $-x^2+24x$

First derivative of this function is given $-2x+24$

The second derivative of the function is given by $-2$.

Putting 1st derivative to zero we get, $-2x+24=0\Rightarrow 2x=24\Rightarrow x=12$

Which means product is maximum only if one number is $12$

then other number is $24-12=12$

NOTE: If sum of two number is constant then product is maximum when both the numbers are equal.