6 The continuous random variable \(X\) has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} \frac { 3 } { 8 } x ^ { 2 } & 0 \leqslant x \leqslant \frac { 1 } { 2 }
\frac { 3 } { 32 } & \frac { 1 } { 2 } \leqslant x \leqslant 11
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Show that:
   1. \(\quad \mathrm { P } \left( X \geqslant 8 \frac { 1 } { 3 } \right) = \frac { 1 } { 4 }\);
   2. \(\quad \mathrm { P } ( X \geqslant 3 ) = \frac { 3 } { 4 }\).
3. Hence write down the exact value of:
   1. the interquartile range of \(X\);
   2. the median, \(m\), of \(X\).
4. Find the exact value of \(\mathrm { P } ( X < m \mid X \geqslant 3 )\).