Answer
This curve is concave up when $\frac{\pi}{2}\lt t \lt \frac{3\pi}{2}$
Work Step by Step
$x = 2sint,$ $y = 3cos t, 0 \lt t \lt 2 \pi$.
Step 1
$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{-3sint}{2cost}=-\frac{3}{2} tan t$
Step 2
Now, find the 2nd derivative:
$\frac{d^{2}y}{dx^{2}}= \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}= \frac{-\frac{3}{2}sec^{2}t}{2cost}=-\frac{3}{4}sec^{3}t$
This curve is concave up when $sec^{3}t \lt0$
If $sec^{3}t \lt 0 $ then $cos t \lt 0$
$\frac{\pi}{2}\lt t \lt \frac{3\pi}{2}$
