Question

Show that area of rectangle of given perimeter is maximum when the rectangle is a square.

Answer 1

Let $x,y$ be the length and breath of the rectangle

$\therefore$ Perimeter is given i.e. is constant

$\therefore$ $2x+2y=2c$ (say)

$x+y=c$

Area of rectangle $A=xy=x(c-x)=cx-x^2$

$\frac{dA}{dx}=c-2x$ for max or min $\frac{dA}{dx}=0\Rightarrow c-2x=0$

$\Rightarrow x=\frac{c}{2}$ and $\Rightarrow y=\frac{c}{2}$

$x=y$ rectangle must be a square.