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$$- \frac{\left(2 \log{\left(x \right)} - 3\right) \sin{\left(\frac{\log{\left(x \right)}}{x} \right)} + \frac{\left(\log{\left(x \right)} - 1\right)^{2} \cos{\left(\frac{\log{\left(x \right)}}{x} \right)}}{x}}{x^{3}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 12331.1389968598$$
$$x_{2} = 8054.84364514281$$
$$x_{3} = 11725.4734391913$$
$$x_{4} = 14440.1068339598$$
$$x_{5} = 11118.2636869515$$
$$x_{6} = 5561.54825670711$$
$$x_{7} = 15638.5343288586$$
$$x_{8} = 9898.77240418325$$
$$x_{9} = 7435.63120139171$$
$$x_{10} = 4297.18474750558$$
$$x_{11} = 13839.1519563589$$
$$x_{12} = 8979.21122848193$$
$$x_{13} = 15339.3438205864$$
$$x_{14} = 6189.18108828368$$
$$x_{15} = 3979.55227274679$$
$$x_{16} = 4930.81779546379$$
$$x_{17} = 5246.56883239806$$
$$x_{18} = 8363.53596375479$$
$$x_{19} = 7745.55030297912$$
$$x_{20} = 10814.0468749925$$
$$x_{21} = 15039.8805513466$$
$$x_{22} = 14139.7810918046$$
$$x_{23} = 13236.9486965209$$
$$x_{24} = 12633.4230016084$$
$$x_{25} = 6813.81282780483$$
$$x_{26} = 3344.44802728503$$
$$x_{27} = 15937.4589075644$$
$$x_{28} = 9286.23820293559$$
$$x_{29} = 10509.4030752904$$
$$x_{30} = 14740.137379608$$
$$x_{31} = 4.20521095380742$$
$$x_{32} = 11422.0679795991$$
$$x_{33} = 12028.4930208733$$
$$x_{34} = 10204.3169835921$$
$$x_{35} = 7125.06075027639$$
$$x_{36} = 5875.74935438494$$
$$x_{37} = 13538.2108318163$$
$$x_{38} = 6501.86133098239$$
$$x_{39} = 8671.65096272814$$
$$x_{40} = 12935.3560739884$$
$$x_{41} = 9592.75219325851$$
$$x_{42} = 4614.32914839737$$
$$x_{43} = 3661.75924307414$$
$$x_{44} = 16236.1240962912$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
True
True
- 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
$$\left(-\infty, 4.20521095380742\right]$$
Convex at the intervals
$$\left[4.20521095380742, \infty\right)$$