Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative$$\frac{\pi^{2} \left(2 - \frac{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1}{\tan^{2}{\left(\frac{\pi x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}{4} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = - \frac{1}{2}$$
$$x_{2} = \frac{1}{2}$$
Сonvexity and concavity intervals:Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left(-\infty, - \frac{1}{2}\right] \cup \left[\frac{1}{2}, \infty\right)$$
Convex at the intervals
$$\left[- \frac{1}{2}, \frac{1}{2}\right]$$