In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative$$\left(\frac{2}{x + 2} - \frac{2 x - 3}{\left(x + 2\right)^{2}}\right) \operatorname{sign}{\left(\frac{2 x - 3}{x + 2} \right)} = 0$$
Solve this equationSolutions are not found,
function may have no extrema