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$$10 x \left(2 x^{2} - 6 x + 3\right) = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 0$$
$$x_{2} = \frac{3}{2} - \frac{\sqrt{3}}{2}$$
$$x_{3} = \frac{\sqrt{3}}{2} + \frac{3}{2}$$
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
$$\left[0, \frac{3}{2} - \frac{\sqrt{3}}{2}\right] \cup \left[\frac{\sqrt{3}}{2} + \frac{3}{2}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right] \cup \left[\frac{3}{2} - \frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2} + \frac{3}{2}\right]$$