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$$\frac{\sqrt{2} \left(- \frac{2}{2 x - 1} + \frac{1}{1 - x}\right)}{4 \sqrt{1 - x} \sqrt{2 x - 1}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = \frac{3}{4}$$
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
$$\left[\frac{3}{4}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{3}{4}\right]$$