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$$\frac{x^{\frac{1}{x}} \left(2 \log{\left(x \right)} - 3 + \frac{\left(\log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 36956.6362249476$$
$$x_{2} = 35842.5039425585$$
$$x_{3} = 43618.932582736$$
$$x_{4} = 48041.0543039226$$
$$x_{5} = 38069.6475342003$$
$$x_{6} = 24632.3562385126$$
$$x_{7} = 52449.4275796145$$
$$x_{8} = 51348.5500563983$$
$$x_{9} = 44725.8295428241$$
$$x_{10} = 26885.164415468$$
$$x_{11} = 4.36777096705602$$
$$x_{12} = 45831.7998080343$$
$$x_{13} = 41402.2587671693$$
$$x_{14} = 33610.7485850034$$
$$x_{15} = 40292.4295040997$$
$$x_{16} = 31374.1183597028$$
$$x_{17} = 30253.89094583$$
$$x_{18} = 29132.3439864666$$
$$x_{19} = 56845.3273080066$$
$$x_{20} = 46936.8671226986$$
$$x_{21} = 28009.4451941101$$
$$x_{22} = 39181.5686882772$$
$$x_{23} = 42511.0842227899$$
$$x_{24} = 55747.4592642109$$
$$x_{25} = 57942.4847067835$$
$$x_{26} = 34727.2189284023$$
$$x_{27} = 49144.3832768866$$
$$x_{28} = 32493.05963509$$
$$x_{29} = 53549.5263991257$$
$$x_{30} = 25759.4750752963$$
$$x_{31} = 54648.8645194423$$
$$x_{32} = 50246.8751110307$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{x^{\frac{1}{x}} \left(2 \log{\left(x \right)} - 3 + \frac{\left(\log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(\frac{x^{\frac{1}{x}} \left(2 \log{\left(x \right)} - 3 + \frac{\left(\log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x^{3}}\right) = 0$$
- 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
$$\left[4.36777096705602, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 4.36777096705602\right]$$