(a)

Consider the length of the rectangle be .

The width of the rectangle be .

The area of the recatngle is .

Then .

The perimeter of the rectangle is .

Substitute in perimeter.

Perimetre of the rectangle is smallest when its first derivative becomes zero.

Apply derivative on each side with respect to .

.

Equate to zero.

The length of the recatngle is .

Substitute in .

The width of the rectangle is .

Here length and width of the recatngle is equal, so it became a square.

(b)

Consider the length of the rectangle be .

The width of the rectangle be .

The perimeter of the rectangle is .

The area of the recatngle is .

Area of the rectangle is greatest when its first derivative becomes zero.

Apply derivative on each side with respect to .

.

Equate to zero.

The length of the recatngle is .

Substitute in .

The width of the rectangle is .

Here length and width of the recatngle is equal, so it became a square.

(a) When perimeter is minimum then it became a square.

(b) When perimeter is minimum then it became a square.