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$$\frac{x \left(\left(2 x - 5\right) \left(\frac{1}{x - 1} + \frac{1}{x - 4}\right) - 2 + \frac{2 x - 5}{x - 1} + \frac{2 x - 5}{x - 4}\right) - 4 x + 10}{\left(x - 4\right)^{2} \left(x - 1\right)^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = - 2 \sqrt[3]{2} - 2^{\frac{2}{3}}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1$$
$$x_{2} = 4$$
$$\lim_{x \to 1^-}\left(\frac{x \left(\left(2 x - 5\right) \left(\frac{1}{x - 1} + \frac{1}{x - 4}\right) - 2 + \frac{2 x - 5}{x - 1} + \frac{2 x - 5}{x - 4}\right) - 4 x + 10}{\left(x - 4\right)^{2} \left(x - 1\right)^{2}}\right) = \infty$$
$$\lim_{x \to 1^+}\left(\frac{x \left(\left(2 x - 5\right) \left(\frac{1}{x - 1} + \frac{1}{x - 4}\right) - 2 + \frac{2 x - 5}{x - 1} + \frac{2 x - 5}{x - 4}\right) - 4 x + 10}{\left(x - 4\right)^{2} \left(x - 1\right)^{2}}\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = 1$$
- is an inflection point
$$\lim_{x \to 4^-}\left(\frac{x \left(\left(2 x - 5\right) \left(\frac{1}{x - 1} + \frac{1}{x - 4}\right) - 2 + \frac{2 x - 5}{x - 1} + \frac{2 x - 5}{x - 4}\right) - 4 x + 10}{\left(x - 4\right)^{2} \left(x - 1\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 4^+}\left(\frac{x \left(\left(2 x - 5\right) \left(\frac{1}{x - 1} + \frac{1}{x - 4}\right) - 2 + \frac{2 x - 5}{x - 1} + \frac{2 x - 5}{x - 4}\right) - 4 x + 10}{\left(x - 4\right)^{2} \left(x - 1\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 4$$
- 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
$$\left[- 2 \sqrt[3]{2} - 2^{\frac{2}{3}}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - 2 \sqrt[3]{2} - 2^{\frac{2}{3}}\right]$$