A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 10−x2. What are the dimensions of such a rectangle with the greatest possible area?
We first formulate everything in mathematical terms: if $\displaystyle x$ is the distance from the origin of one lower corner of the rectangle, then the base of the rectangle is $\displaystyle 2x$, and the height of the rectangle is $\displaystyle 10-x^2$. Its area is therefore
$\displaystyle A=2x\cdot(10-x^2).$
To maximize $\displaystyle A$, we remember that $\displaystyle A$ is differentiable everywhere and approaches $\displaystyle 0$ at the boundary points of its domain, and therefore that $\displaystyle A$ will attain an extremum at some point at which $\displaystyle \frac{dA}{dx}=0$.