## Convexity and concavity of a function.

Definition 6.10.5   Let be a function defined over an interval .
(i)
The function is convex on if, for any two points and on the graph, the arc is below the chord (see Figure 9(a)).
(ii)
The function is concave on if, for any two points and on the graph, the chord is below the arc (see Figure 9(b)).

Proposition 6.10.6   Let be a function defined and differentiable, at least twice, on an interval .
(i)
If for every , the function is convex on ;
(ii)
If for every , the function is concave on .

Example 6.10.7   Take . The function is a polynomial function, thus it is differentiable over . We have:

Study the sign of the second derivative:
• , i.e. is concave over the interval ;
• , i.e. is convex over the interval .
The graph of is displayed on Figure 10.

Definition 6.10.8 (See Figure 11)   Let be a function, defined on an interval , and let be an interior point of . Suppose that convexity and concavity interchange at . The we say that is a point of inflection of .

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 .

Example 6.10.9
1. If , then has an inflection point at 0.
2. The sine function has an inflection point at each .
3. The cosine function has an inflection point at each .
4. The exponential function has no inflection point.

Remark 6.10.10   Denote by the graph of the function and by the point whose coordinate are .

• 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 .
See Figure 12.

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