Solutions

The perimeter of any rectangle can be expressed as

$$p = 2m + 2n$$

for $m, n \in \mathbb{R}$. As a result,

$$m = \frac{p}{2} - n.$$

It follows that the area of a rectangle so defined is

$$a = mn = \left(\frac{p}{2} - n\right)n = -n^2 + \frac{p}{2}n$$

and it has a global maximum because it is a polynomial of even degree and the leading coefficient is negative. Its critical point is

$$\begin{align*} \frac{da}{dn} = 0 &\implies -2n + \frac{p}{2} = 0 \\ &\implies n = \frac{p}{4}. \end{align*}$$

Plugging this back into the formula of areas, we get

$$\begin{align*} mn = m\frac{p}{4} &\implies mn = m\left(\frac{m+n}{2}\right) \\ &\implies 2mn = m^2 + mn \\ &\implies mn = m^2 \\ &\implies n = m, \end{align*}$$

assuming that $m \le 0$ is inadmissible.