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$\displaystyle 2x + 2y = 100$
$\displaystyle x + y = 50$
Formula for the area is, $\displaystyle A = xy$.
As $\displaystyle x +y = 50$, $\displaystyle y = 50 -x$
So our formula becomes, $\displaystyle A = x (50-x) = 50x - x^2$.
To find the extremum point, we will differentiate it and $\displaystyle A' = 0$ will give us the extremum point.
$\displaystyle A = 50x - x^2$
$\displaystyle A' = 50 - 2x$
$\displaystyle A' = 50 - 2x = 0$
$\displaystyle x = 25$
There's an extremum point at $\displaystyle x = 25$, but we don't know whether it's a minima or a maxima. The second derivative test will help us to determine what it is
If $\displaystyle A''(25) > 0 $, then it's a minima. If $\displaystyle A''(25) < 0$, it's a maxima.
$\displaystyle A = 50 x - x^2$
$\displaystyle A' = 50 - 2x$
$\displaystyle A'' = -2$
$\displaystyle A''(25) = -2$, which is less than 0 and which makes our point a maxima
So $\displaystyle x = 25$ will give us the maximum area.