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$$4^{\log{\left(2 \cos{\left(x \right)} \right)}} \left(- \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\log{\left(4 \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 1\right) \log{\left(4 \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = - 2 \operatorname{atan}{\left(\sqrt{-1 + 2 \sqrt{2} \sqrt{-1 + \log{\left(4 \right)}} \sqrt{\log{\left(2 \right)}} + \log{\left(16 \right)}} \right)}$$
$$x_{2} = 2 \operatorname{atan}{\left(\sqrt{-1 + 2 \sqrt{2} \sqrt{-1 + \log{\left(4 \right)}} \sqrt{\log{\left(2 \right)}} + \log{\left(16 \right)}} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{-1 + \log{\left(4 \right)}} \sqrt{\log{\left(2 \right)}} - 1 + \log{\left(16 \right)}} \right)}$$
$$x_{4} = 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{-1 + \log{\left(4 \right)}} \sqrt{\log{\left(2 \right)}} - 1 + \log{\left(16 \right)}} \right)}$$
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
$$\left[2 \operatorname{atan}{\left(\sqrt{-1 + 2 \sqrt{2} \sqrt{-1 + \log{\left(4 \right)}} \sqrt{\log{\left(2 \right)}} + \log{\left(16 \right)}} \right)}, \infty\right)$$
Convex at the intervals
$$\left[- 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{-1 + \log{\left(4 \right)}} \sqrt{\log{\left(2 \right)}} - 1 + \log{\left(16 \right)}} \right)}, 2 \operatorname{atan}{\left(\sqrt{- 2 \sqrt{2} \sqrt{-1 + \log{\left(4 \right)}} \sqrt{\log{\left(2 \right)}} - 1 + \log{\left(16 \right)}} \right)}\right]$$