a 10 ft display sign is 10 feet off the ground. Find the distance from the sign of a point on the ground where the subtended angle of the sign is a maximum.
Let $\displaystyle x$ be the optimal distance from the sign, $\displaystyle \alpha$ be the subtended angle of the sign and $\displaystyle \beta$ be the angle formed by the ground and the ray from the optimal viewpoint to the bottom of the sign. Then $\displaystyle \tan\beta=10/x$ and $\displaystyle \tan(\alpha+\beta)=20/x$. Using the formula for the tangent of difference, you can express $\displaystyle \tan((\alpha+\beta)-\beta)=\tan\alpha$ through $\displaystyle x$. Since $\displaystyle \tan$ is a monotonic function on $\displaystyle [0,\pi/2)$, the maximum point of $\displaystyle \tan(\alpha(x))$ is also the maximum point of $\displaystyle \alpha(x)$.