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$$\frac{2 \left(\frac{x}{x - 4} - 1\right) \left(\frac{1}{x - 4} + \frac{1}{x}\right)}{x} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 2$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 4$$
$$\lim_{x \to 4^-}\left(\frac{2 \left(\frac{x}{x - 4} - 1\right) \left(\frac{1}{x - 4} + \frac{1}{x}\right)}{x}\right) = \infty$$
$$\lim_{x \to 4^+}\left(\frac{2 \left(\frac{x}{x - 4} - 1\right) \left(\frac{1}{x - 4} + \frac{1}{x}\right)}{x}\right) = \infty$$
- limits are equal, then skip the corresponding 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
$$\left[2, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2\right]$$