Question

If two real numbers $x$ and $y$ such that $x+y=12$ then find the maximum value of their product.

Answer 1

Given for two real numbers $x,y$ $x+y=12$......(1).

Let $p=xy$

or, $p=x(12-x)$.......(2) [Using (1)]

Now $\frac{dp}{dx}=12-2x$.

And $\frac{d^{2}p}{d{x}^2}=-2$.

Now for maximum or minimum value of $p$ we must have,

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

or, $12-2x=0$

or, $x=6$, this gives $y=6$. [ From (1)].

Now for these value of $x$ and $y$ we have $\frac{d^{2}p}{d{x}^2}<0$.

So for $x=6,y=6$ we have maximum value of $p$.

$\therefore$ the maximum value of $p$ is $6\times 6=36$.