3 The continuous random variable $X$ has probability density function f given by

$$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x < 2 \\ k ( 6 - x ) & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$

where $k$ is a constant.\\
(a) Show that $k = \frac { 3 } { 40 }$.\\

(b) Given that $\mathrm { E } ( X ) = 2.5$, find $\operatorname { Var } ( X )$.\\

(c) Find the median value of $X$.\\

\hfill \mbox{\textit{CAIE Further Paper 4 2022 Q3}}