Graphing y = 9^(1/(x-3))

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
$$\frac{9^{\frac{1}{x - 3}} \left(2 + \frac{\log{\left(9 \right)}}{x - 3}\right) \log{\left(9 \right)}}{\left(x - 3\right)^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 3 - \log{\left(3 \right)}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 3$$

$$\lim_{x \to 3^-}\left(\frac{9^{\frac{1}{x - 3}} \left(2 + \frac{\log{\left(9 \right)}}{x - 3}\right) \log{\left(9 \right)}}{\left(x - 3\right)^{3}}\right) = 0$$
$$\lim_{x \to 3^+}\left(\frac{9^{\frac{1}{x - 3}} \left(2 + \frac{\log{\left(9 \right)}}{x - 3}\right) \log{\left(9 \right)}}{\left(x - 3\right)^{3}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 3$$
- 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
$$\left[3 - \log{\left(3 \right)}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 3 - \log{\left(3 \right)}\right]$$