Chapter 5, Problem 5.8TE

Let X be a random variable that takes on values between 0 and c. That is, $p\left\{0\le X\le c\right\}=1$ .Show that $\mathrm{var}\left(X\right)\le \frac{c^2}{4}$

Hint: One approach is to first argue that $E\left[X^2\right]\le cE\left[X\right]$ and then use this inequality to show that $\mathrm{var}\left(X\right)\le c^2\left[a\left(1-a\right)\right]$ where $a=\frac{E\left[X\right]}{c}$