The left-hand side will always be negative. Therefore the inflection points must be where the tangent is negative. One each in the ranges $-\frac{3\pi}{2} < x^2 < -\pi$, $-\frac{\pi}{2} < x^2 < 0$, $\frac{\pi}{2} < x^2 < \pi$, and $\frac{3\pi}{2} < x^2 < 2\pi$.

Since you're graphing, you only need approximate values of $x$. Might I suggest the Newton-Raphson method?

Start with a guess for $x$ in the desired range, then update the guess using

$\displaystyle x_{new} = x_{previous} - \frac{y''(x_{previous})}{y'''(x_{previous})}$.

It should only take a couple iterations to get a good approximation for $x$ for each range.