Using derivatives to find point of inflexion

[soccerboi]

How do you test to see if a point is a point of inflexion? I know that you test a point to the left of it and right of it.
Is it correct to say: A point of inflexion occurs (by testing points on either side) when the derivative doesn't change signs an the second derivative changes sign?

[DisaFear]

Indeed. If you look at the graph below, the gradient has the same sign on either side. The concavity changes, though, on either side, which the second derivative can tell you all about.

If f''(x)>0 over an interval, then the function f' is increasing over that interval, so the graph of f is concave up over that interval
And vice versa.

[Mao]

, , and

Alternatively, show that and first becomes non-zero for an odd