calculus - Why do second or higher derivatives work for finding concavity and inflection points?

Say we have the function $f(x)=(x-2)^3+3$, whose graph is

and we want to find at what regions does $f$ have a positive/negative concavity, and where the inflection points are.

I learned to answer these questions doing:
$$\begin{align*} f^{'}(x) &= 3(x-2)^2 \\ f^{''}(x) &= 2\cdot 3(x-2)^1 \\ f^{''}(x) &= 0 \implies x = 2 \\ f(2)&= 3 \end{align*}$$
$\therefore$
Concavity is positive within $(2, \infty)$, negative within $(- \infty, 2)$
Inflection point(s): $(2, 3)$

But why does this work? Will I have issues when the function has multiple inflection points or do I just have to be more careful? And what if the degree of a function were very high, say of degree 6? Would I have to keep computing the derivative until I get a derivative of degree 1 or does it only take until the second derivative?