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$$\frac{- \left(\frac{\log{\left(x \right)}}{- x \log{\left(x \right)} + x + 3} - \frac{1}{x}\right) \left(- 2 x \log{\left(x \right)} + x + 3\right) - \frac{\left(- 2 x \log{\left(x \right)} + x + 3\right) \log{\left(x \right)}}{- x \log{\left(x \right)} + x + 3} + 2 \log{\left(x \right)} + 1 + \frac{- 2 x \log{\left(x \right)} + x + 3}{x}}{x^{2} \left(- x \log{\left(x \right)} + x + 3\right)^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 7672.99190661819$$
$$x_{2} = 8593.40594449503$$
$$x_{3} = 4202.73882370202$$
$$x_{4} = 6205.89669284138$$
$$x_{5} = 3119.08215025949$$
$$x_{6} = 7121.97119628465$$
$$x_{7} = 6571.96316874159$$
$$x_{8} = 1686.41288243452$$
$$x_{9} = 6023.05584181322$$
$$x_{10} = 3840.71109908618$$
$$x_{11} = 2043.32624407698$$
$$x_{12} = 3479.47473740271$$
$$x_{13} = 8409.13015490223$$
$$x_{14} = 3659.99069733403$$
$$x_{15} = 2401.00158869569$$
$$x_{16} = 8777.7736938764$$
$$x_{17} = 5293.06907088743$$
$$x_{18} = 1033.83458747321$$
$$x_{19} = 1864.79216386676$$
$$x_{20} = 4384.03319415703$$
$$x_{21} = 7856.87695934012$$
$$x_{22} = 2222.05541163804$$
$$x_{23} = 5475.35060413016$$
$$x_{24} = 8040.86332643922$$
$$x_{25} = 2759.58102657328$$
$$x_{26} = 1151.17705697708$$
$$x_{27} = 8224.94851986206$$
$$x_{28} = 4747.15195156773$$
$$x_{29} = 7489.2107657662$$
$$x_{30} = 4565.50623012315$$
$$x_{31} = 2580.17551826721$$
$$x_{32} = 1329.77983397393$$
$$x_{33} = 2939.21762712816$$
$$x_{34} = 7305.53625021851$$
$$x_{35} = 6938.51857053392$$
$$x_{36} = 4928.96461937267$$
$$x_{37} = 5840.34842648046$$
$$x_{38} = 1508.1171708508$$
$$x_{39} = 5110.93874116986$$
$$x_{40} = 5657.77857087083$$
$$x_{41} = 3299.16978222964$$
$$x_{42} = 6755.18147764661$$
$$x_{43} = 4021.62932451542$$
$$x_{44} = 6388.86705018131$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$x_{2} = 4.97062575954423$$
$$\lim_{x \to 0^-}\left(\frac{- \left(\frac{\log{\left(x \right)}}{- x \log{\left(x \right)} + x + 3} - \frac{1}{x}\right) \left(- 2 x \log{\left(x \right)} + x + 3\right) - \frac{\left(- 2 x \log{\left(x \right)} + x + 3\right) \log{\left(x \right)}}{- x \log{\left(x \right)} + x + 3} + 2 \log{\left(x \right)} + 1 + \frac{- 2 x \log{\left(x \right)} + x + 3}{x}}{x^{2} \left(- x \log{\left(x \right)} + x + 3\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{- \left(\frac{\log{\left(x \right)}}{- x \log{\left(x \right)} + x + 3} - \frac{1}{x}\right) \left(- 2 x \log{\left(x \right)} + x + 3\right) - \frac{\left(- 2 x \log{\left(x \right)} + x + 3\right) \log{\left(x \right)}}{- x \log{\left(x \right)} + x + 3} + 2 \log{\left(x \right)} + 1 + \frac{- 2 x \log{\left(x \right)} + x + 3}{x}}{x^{2} \left(- x \log{\left(x \right)} + x + 3\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
$$\lim_{x \to 4.97062575954423^-}\left(\frac{- \left(\frac{\log{\left(x \right)}}{- x \log{\left(x \right)} + x + 3} - \frac{1}{x}\right) \left(- 2 x \log{\left(x \right)} + x + 3\right) - \frac{\left(- 2 x \log{\left(x \right)} + x + 3\right) \log{\left(x \right)}}{- x \log{\left(x \right)} + x + 3} + 2 \log{\left(x \right)} + 1 + \frac{- 2 x \log{\left(x \right)} + x + 3}{x}}{x^{2} \left(- x \log{\left(x \right)} + x + 3\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 4.97062575954423^+}\left(\frac{- \left(\frac{\log{\left(x \right)}}{- x \log{\left(x \right)} + x + 3} - \frac{1}{x}\right) \left(- 2 x \log{\left(x \right)} + x + 3\right) - \frac{\left(- 2 x \log{\left(x \right)} + x + 3\right) \log{\left(x \right)}}{- x \log{\left(x \right)} + x + 3} + 2 \log{\left(x \right)} + 1 + \frac{- 2 x \log{\left(x \right)} + x + 3}{x}}{x^{2} \left(- x \log{\left(x \right)} + x + 3\right)^{2}}\right) = -\infty$$
- 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:
Have no bends at the whole real axis