4. The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 0
k \left( x ^ { 3 } - \frac { 3 } { 8 } x ^ { 4 } \right) & 0 < x \leqslant 2
1 & x > 2 \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 2 }\)
2. Showing your working clearly, use calculus to find
   1. \(\mathrm { E } ( X )\)
   2. the mode of \(X\)