7 The continuous random variable \(X\) has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 16 } x ^ { 3 } & 0 \leqslant x \leqslant 2
\frac { 1 } { 6 } ( 5 - x ) & 2 \leqslant x \leqslant 5
0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f.
2. Prove that the cumulative distribution function of \(X\) for \(2 \leqslant x \leqslant 5\) can be written in the form $$\mathrm { F } ( x ) = 1 - \frac { 1 } { 12 } ( 5 - x ) ^ { 2 }$$
3. Hence, or otherwise, determine \(\mathrm { P } ( X \geqslant 3 \mid X \leqslant 4 )\).