- (i)
- If for every , the function is convex on ;
- (ii)
- If for every , the function is concave on .

Study the sign of the second derivative:

- , i.e. is concave over the interval ;
- , i.e. is convex over the interval .

According to Proposition 10.6, if is (at least) twice differentiable in a neighborhood of , then is an inflection point of provided the following conditions are fulfilled:

- ;
- the sign of changes at .

- If , then has an inflection point at 0.
- The sine function has an inflection point at each .
- The cosine function has an inflection point at each .
- The exponential function has no inflection point.

- If the function is concave at a point , then the tangent to at is over in a neighborhood of .
- If the function is convex at a point , then the tangent to at is under in a neighborhood of .

If is a point of inflection of a function , then at the point , the graph of passes through its tangent (see Figure 11).

Noah Dana-Picard 2007-12-28