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$$- \frac{2 \left(\frac{4 x \left(x - 2\right)}{- x^{2} + 4 x + 32} + \frac{\left(x^{2} - 16\right) \left(\frac{4 \left(x - 2\right)^{2}}{- x^{2} + 4 x + 32} + 1\right)}{- x^{2} + 4 x + 32} + 1\right)}{- x^{2} + 4 x + 32} = 0$$
Solve this equationSolutions are not found,
maybe, the function has no inflections