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$$\left(- \frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} + 4 \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)}}{4} + \frac{1}{4}\right) = 0$$
Solve this equationThe roots of this equation
$$x_{1} = - \frac{\pi}{6}$$
$$x_{2} = \frac{\pi}{6}$$
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
$$\left[\frac{\pi}{6}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{\pi}{6}\right]$$