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$$\frac{- \frac{1}{\left(2 x - 1\right)^{\frac{3}{2}}} - \frac{2}{x \sqrt{2 x - 1}} + \frac{2 \sqrt{2 x - 1}}{x^{2}}}{x} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 1 - \frac{\sqrt{3}}{3}$$
$$x_{2} = \frac{\sqrt{3}}{3} + 1$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \frac{1}{\left(2 x - 1\right)^{\frac{3}{2}}} - \frac{2}{x \sqrt{2 x - 1}} + \frac{2 \sqrt{2 x - 1}}{x^{2}}}{x}\right) = - \infty i$$
$$\lim_{x \to 0^+}\left(\frac{- \frac{1}{\left(2 x - 1\right)^{\frac{3}{2}}} - \frac{2}{x \sqrt{2 x - 1}} + \frac{2 \sqrt{2 x - 1}}{x^{2}}}{x}\right) = \infty i$$
- 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
$$\left[\frac{\sqrt{3}}{3} + 1, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{\sqrt{3}}{3} + 1\right]$$