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$$\frac{\frac{2 \sqrt{x}}{\left(x + 100\right)^{2}} - \frac{1}{\sqrt{x} \left(x + 100\right)} - \frac{1}{4 x^{\frac{3}{2}}}}{x + 100} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = - \frac{200 \sqrt{3}}{3} + 100$$
$$x_{2} = 100 + \frac{200 \sqrt{3}}{3}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -100$$
$$\lim_{x \to -100^-}\left(\frac{\frac{2 \sqrt{x}}{\left(x + 100\right)^{2}} - \frac{1}{\sqrt{x} \left(x + 100\right)} - \frac{1}{4 x^{\frac{3}{2}}}}{x + 100}\right) = - \infty i$$
Let's take the limit$$\lim_{x \to -100^+}\left(\frac{\frac{2 \sqrt{x}}{\left(x + 100\right)^{2}} - \frac{1}{\sqrt{x} \left(x + 100\right)} - \frac{1}{4 x^{\frac{3}{2}}}}{x + 100}\right) = \infty i$$
Let's take the limit- the limits are not equal, so
$$x_{1} = -100$$
- 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:
Have no bends at the whole real axis