Solutions - 0260-03

First derivative:

$$\begin{array}{c}\frac{dy}{dx}=\frac{y^{\prime }(t)}{x^{\prime }(t)}\\ \\ \gg \gg \frac{3t^2-4}{2t}\gg \gg \frac{3}{2}t-2t^{-1}\end{array}$$

Second derivative:

$$\begin{array}{c}\frac{d^{2}y}{d{x}^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot \frac{1}{x^{\prime }(t)}\\ \\ \gg \gg \frac{d}{dt}(\frac{3}{2}t-2t^{-1})\cdot \frac{1}{2t}\\ \\ \gg \gg \frac{3}{4t}+\frac{1}{t^3}\gg \gg \frac{3t^2+4}{4t^3}\end{array}$$

This is positive if-and-only-if $t>0$. (Numerator always positive, denominator same sign as $t$.) So:

$$\text{Concave up:}t\in (0,+\infty )$$