Question

Find two number, such that their product is $64$ and their sum is minimum.

Answer 1

Let the numbers be $x$ and $y$.

Then by the problem,

$xy=64$

or, $y=\frac{64}{x}$........(1).

We are to minimize the sum of the two numbers.

Let $S=x+y$

or, $S=x+\frac{64}{x}$ [Using (1)]

Now $\frac{dS}{dx}=1-\frac{64}{x^2}$

Again

$\frac{d^{2}S}{d{x}^2}=2\frac{64}{x^3}$.......(2).

For maximum or minimum value of $S$ we must have

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

or, $1-\frac{64}{x^2}=0$

or, $x^2=64$

or, $x=8,-8$.

Now it is clear that ${\frac{d^S}{d{x}^2}|}_{x=8}=\frac{1}{4}>0$

So minimum value attained at $x=8$.

Now $x=8$ gives $y=8$.

So the numbers are $8,8$.