Question

Determine two positive numbers whose sum is 24 and whose product is maximum.

Answer 1

Let the two number is $x$ and $y$

According to given question.

Sum of two number $=24$

$x+y=24$

Then, $y=24-x.......\left(1\right)$

Again, product

$P=\mathrm{maximum}=x\times y$

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

$P=24x-24x^2.......\left(2\right)$

We know that,

For maximum value

$\frac{dP}{dx}=0$

Now differentiating equation (2) with respect to $x$ and we get,

$\frac{dp}{dx}=24-2x$

$24-2x=0$

$2x=24$

$x=12$

Again, differentiating equation (2) with respect to $x$ and we get,

$\frac{d^{2}p}{d{x}^2}=-2<0$

Then,$P$ is minimum at point $x=12$

Hence, the required numbers is $x=12$ and $y=24-x=24-12=12$ $i.e.(12,12)$

This is the solution.