6 The continuous random variable \(X\) has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( x ^ { 2 } + 1 \right) & 0 \leqslant x \leqslant 1
\frac { 1 } { 4 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$

1. The cumulative distribution function of \(X\) is denoted by \(\mathrm { F } ( x )\). Show that, for \(0 \leqslant x \leqslant 1\), $$\mathrm { F } ( x ) = \frac { 1 } { 8 } x \left( x ^ { 2 } + 3 \right)$$
2. Hence, or otherwise, verify that the median value of \(X\) is 1 .
3. Show that the upper quartile, \(q\), satisfies the equation \(q ^ { 2 } - 5 q + 5 = 0\) and hence that \(q = \frac { 1 } { 2 } ( 5 - \sqrt { 5 } )\).
4. Calculate the exact value of \(\mathrm { P } ( q < X < 1.5 )\).