Module 4 : Local / Global Maximum / Minimum and  Curve Sketching
Lecture 11 : Convex / Concave Functions [Section 11.2]
 

For
                  

Thus,  Hence, is strictly concave down for and concave up for .It has a point of inflection at

Theorem 11.2.9 suggests that if for a function exists at a point of inflection , then . That this is indeed the case in proved in the next corollary.

  Click here to see a visualization : Applet 11.1
11.2.11 Corollary:
 

Let

(i)
 Necessary condition for points of inflection:
(ii) 
 Third derivative test for point of inflection:
 

 
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