What is the largest possible area for a triangle whose hypotenuse is 5cm. long?
Since it has a hypotenuse, we'll assume that it's right-angled. Call the base of the triangle $\displaystyle \displaystyle \begin{align*} x \end{align*}$, then the height must be $\displaystyle \displaystyle \begin{align*} \sqrt{25 - x^2} \end{align*}$ by Pythagoras.
So the area is $\displaystyle \displaystyle \begin{align*} \frac{1}{2} x\,\sqrt{25 - x^2} \end{align*}$. Differentiate this, set the derivative equal to 0, solve for x, then evaluate the area at this value of x.