Problem

The curve $C$ has parametric equations

$$x=\frac{2}{3}t^{\frac{3}{2}}-2t^{\frac{1}{2}},\qquad y=2t+5,\qquad\text{for }0<t\le3.$$

(a) Find the exact length of $C$.

(b) Find the set of values of $t$ for which

$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}>0.$$