a) The CDF is found by taking the integral of the density starting at negative infinity.Let F(x) be the CDF of X. Based on the given information, $F(x) = 0$ for $x \le 0$.
For $x \ge 0$, $F(x)$ can be found as the following

$$\begin{aligned}F(x)&= \int_{-\infty}^x \frac{u}{2}du \\&= \int_{-\infty}^0 f(u)du + \int_{0}^x f(u)du \\&= 0 + \int_{0}^{x}\frac{u}{2}du\\&= \frac{x^2}{4}\end{aligned}$$

(b) $$\begin{aligned}P(X<1) = F(1) = \frac{1^2}{4} = \frac{1}{4}\end{aligned}$$

(c) $$\begin{aligned}P(X>\frac{3}{2}) &= 1 - P(X \le \frac{3}{2}) \\&= 1-F(\frac{3}{2})\\&= 1-\frac{9}{16} = \frac{7}{16}\end{aligned}$$