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$$\frac{x^{- \frac{\log{\left(x \right)} - 1}{x}} \left(- \left(2 \log{\left(x \right)} - 5\right) \log{\left(x \right)} + 3 \log{\left(x \right)} - 5 + \frac{\left(- \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + \log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 77931.734784807$$
$$x_{2} = 102935.111650972$$
$$x_{3} = 97150.494187098$$
$$x_{4} = 95722.5225642332$$
$$x_{5} = 25.8035786397959$$
$$x_{6} = 94304.3504508803$$
$$x_{7} = 108794.49750591$$
$$x_{8} = 87961.4823832922$$
$$x_{9} = 100030.394442753$$
$$x_{10} = 104394.661040139$$
$$x_{11} = 78739.57382288$$
$$x_{12} = 105858.105008557$$
$$x_{13} = 80261.0338703568$$
$$x_{14} = 91504.4658766876$$
$$x_{15} = 81303.7826563622$$
$$x_{16} = 101480.105777427$$
$$x_{17} = 107324.880569688$$
$$x_{18} = 84848.3886498729$$
$$x_{19} = 88768.8309162064$$
$$x_{20} = 98586.8506610603$$
$$x_{21} = 77784.2394803298$$
$$x_{22} = 77627.743626596$$
$$x_{23} = 83612.7146571421$$
$$x_{24} = 79323.6060511737$$
$$x_{25} = 90127.2011741679$$
$$x_{26} = 92897.6599631321$$
$$x_{27} = 86124.2767724825$$
$$x_{28} = 82426.889855648$$
$$x_{29} = 78528.2363867842$$
$$x_{30} = 87433.0062814698$$
$$x_{31} = 2.08138681592069$$
$$x_{32} = 111724.750004227$$
$$x_{33} = 81299.2959882933$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{x^{- \frac{\log{\left(x \right)} - 1}{x}} \left(- \left(2 \log{\left(x \right)} - 5\right) \log{\left(x \right)} + 3 \log{\left(x \right)} - 5 + \frac{\left(- \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + \log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(\frac{x^{- \frac{\log{\left(x \right)} - 1}{x}} \left(- \left(2 \log{\left(x \right)} - 5\right) \log{\left(x \right)} + 3 \log{\left(x \right)} - 5 + \frac{\left(- \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + \log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}}\right) = 0$$
- the limits are not equal, so
$$x_{1} = 0$$
- 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
$$\left[2.08138681592069, 25.8035786397959\right]$$
Convex at the intervals
$$\left(-\infty, 2.08138681592069\right] \cup \left[25.8035786397959, \infty\right)$$