Problem

The diagram shows the continuous random variable $X$ with probability density function $f$ given by

$$f(x)= \begin{cases} \frac{1}{128}(4ax-bx^3), & 0 \le x \le 4,\\ c, & 4 \le x \le 6,\\ 0, & \text{otherwise}, \end{cases}$$

where $a$, $b$ and $c$ are constants.

The upper quartile of $X$ is equal to 4.

(a) Show that $c=\frac{1}{8}$ and find the values of $a$ and $b$.

(b) Find the exact value of the median of $X$.

(c) Find $\operatorname{E}(\sqrt{X})$, giving your answer correct to 2 decimal places.