The continuous random variable \(X\) has cumulative distribution function F given by
\(F(x)=\left\{\begin{array}{lc} 0 & x\lt -1, \\ \frac{1}{2}(1+x)^{2} & -1 \leqslant x \leqslant 0, \\ 1-\frac{1}{2}(1-x)^{2} & 0\lt x \leqslant 1, \\ 1 & x\gt 1 . \end{array}\right.\)
(a) Find the probability density function of \(X\).

(b) Find \(\mathrm{P}\left(-\frac{1}{2} \leqslant X \leqslant \frac{1}{2}\right)\).

(c) Find \(\mathrm{E}\left(X^{2}\right)\).

(d) Find \(\operatorname{Var}\left(X^{2}\right)\).