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?
By definition, a function $f(x)$ is concave on $[a,b]$ if, for every $x,y\in[a,b]$ and for any $\alpha\in[0,1]$, $f((1-\alpha)x+\alpha y)\geq (1-\alpha)f(x)+\alpha f(y)$. This simply means that any point chosen between $x$ and $y$ is at or above the straight line connecting $(x,f(x))$ and $(y,f(y))$. The first derivative measures the instantaneous slope at a point. The second derivative is the derivative of the first derivative, and it measures concavity and can thus be used to determine points of inflection (which occur when the concavity changes sign). See here and this site for more info.