Think of various rectangles with perimeter 40 units (the sides need not be integers).

(A) How many such rectangles are there?

(B) Is there one of largest area? What are its dimensions?

(C) Is there one of smallest area? Do either of these answers surprise you?                                       [Page No. 134]

A perimeter of $40$ means $2(\mathrm{length} + \mathrm{width}) = 40$, so $\mathrm{length} + \mathrm{width} = 20$.
If one side is $x$, the other is $(20 - x)$, with $0 < x < 20$, and the area is $x(20 - x)$.

(A) Infinitely many. The side $x$ can take any value between $0$ and $20$ such as $1 \times 19$, $5 \times 15$, $9.5 \times 10.5$,
and so on, giving endlessly many rectangles.

(B) $\text{Area} = x(20 - x) = 20x - x^2$, which is greatest when $x = 10$.
The largest-area rectangle is therefore the $10 \times 10$ square, with area $100$ sq units.
(Among all rectangles of a fixed perimeter, the square has the greatest area.)

(C) There is no smallest. As $x$ approaches $0$ (or $20$), the rectangle becomes a long, thin sliver and its area shrinks towards $0$, but it never reaches a least positive value.
So, the area can be made as small as we like, yet no rectangle is the smallest.

The surprise is that there is a clear largest rectangle (the square) but no smallest one, a neat reminder that a quantity can have a maximum without having a minimum.