Graphing y = log(3*x)/x

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
$$\frac{2 \log{\left(3 x \right)} - 3}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{e^{\frac{3}{2}}}{3}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$

$$\lim_{x \to 0^-}\left(\frac{2 \log{\left(3 x \right)} - 3}{x^{3}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \log{\left(3 x \right)} - 3}{x^{3}}\right) = -\infty$$
- 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
$$\left[\frac{e^{\frac{3}{2}}}{3}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{e^{\frac{3}{2}}}{3}\right]$$