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$$\frac{2 \left(- 2 x + \left(x + 2\right) \left(\frac{4 x^{2}}{x^{2} - 9} - 1\right)\right)}{\left(x^{2} - 9\right)^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -2 - \sqrt[3]{5} + 5^{\frac{2}{3}}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -3$$
$$x_{2} = 3$$
$$\lim_{x \to -3^-}\left(\frac{2 \left(- 2 x + \left(x + 2\right) \left(\frac{4 x^{2}}{x^{2} - 9} - 1\right)\right)}{\left(x^{2} - 9\right)^{2}}\right) = -\infty$$
$$\lim_{x \to -3^+}\left(\frac{2 \left(- 2 x + \left(x + 2\right) \left(\frac{4 x^{2}}{x^{2} - 9} - 1\right)\right)}{\left(x^{2} - 9\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -3$$
- is an inflection point
$$\lim_{x \to 3^-}\left(\frac{2 \left(- 2 x + \left(x + 2\right) \left(\frac{4 x^{2}}{x^{2} - 9} - 1\right)\right)}{\left(x^{2} - 9\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 3^+}\left(\frac{2 \left(- 2 x + \left(x + 2\right) \left(\frac{4 x^{2}}{x^{2} - 9} - 1\right)\right)}{\left(x^{2} - 9\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 3$$
- is an inflection point
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
$$\left(-\infty, -2 - \sqrt[3]{5} + 5^{\frac{2}{3}}\right]$$
Convex at the intervals
$$\left[-2 - \sqrt[3]{5} + 5^{\frac{2}{3}}, \infty\right)$$