4
\includegraphics[max width=\textwidth, alt={}, center]{a9f9cf66-0734-4316-99ae-c57090d08135-08_353_1141_255_463}
The diagram shows the continuous random variable \(X\) with probability density function f given by
$$f ( x ) = \begin{cases} \frac { 1 } { 128 } \left( 4 a x - b x ^ { 3 } \right) & 0 \leqslant x \leqslant 4
c & 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$
where \(a , b\) and \(c\) are constants.
The upper quartile of \(X\) is equal to 4 .
- Show that \(c = \frac { 1 } { 8 }\) and find the values of \(a\) and \(b\).
- Find the exact value of the median of \(X\).
- Find \(\mathrm { E } ( \sqrt { X } )\), giving your answer correct to 2 decimal places.