3 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } x \mathrm { e } ^ { - \frac { 1 } { 2 } x } & x \geqslant 0
0 & \text { otherwise } . \end{cases}$$

1. Show that the moment generating function of \(X\) is \(( 1 - 2 t ) ^ { - 2 }\) for \(t < \frac { 1 } { 2 }\), and state why the condition \(t < \frac { 1 } { 2 }\) is necessary.
2. Use the moment generating function to find \(\operatorname { Var } ( X )\).