Show that all the rectangles with a given perimeter, the square has the largest area.

Explanation

Let x and y be length and breadth

And A and P be area and perimeter

`A=x cdot y text { and } P=2(x+y) `

`Rightarrow A=x(frac{P}{2}-x) (because y=frac{P}{2}-x) `

`Rightarrow A =frac{P}{2} x-x^{2} quad Rightarrow frac{d A}{d x}=frac{P}{2}-2 x`

For maximum and minimum of `A`.

`Rightarrow quad frac{P}{2}-2 x=0 quad Rightarrow quad x=frac{P}{4}`

Again `frac{d^{2} A}{d x^{2}}=-2`

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