1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

$$\mathrm { f } ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 4
b x + c & 4 < x \leqslant d
0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\), \(c\) and \(d\) are constants such that

• \(b x + c = a x ^ { 3 }\) at \(x = 4\)
• \(b x + c\) is a straight line segment with end coordinates ( \(4,64 a\) ) and ( \(d , 0\) )
  1. State the mode of \(X\)

Given that the mode of \(X\) is equal to the median of \(X\)

use algebraic integration to show that \(a = \frac { 1 } { 128 }\)

Find the value of \(d\)

Hence find the value of \(b\) and the value of \(c\)