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

Consider

The function is differentiable everywhere with

Since the derivative function is strictly increasing, the function is always concave up.

11.2.8 Theorem (Second derivative test for convexity / concavity):
  Let be such that exists in . Then the following hold:
(i)

concave upward if and only if for all .

(ii)
 concave downward if and only if for all .
(iii)
 If for all , then is strictly concave upward.
(iv)
 If for all , then is strictly concave downward.
 
11.2.9 Theorem (Tests for point of inflection):
  Let
(i)

Let exist in , for some , except possibly at , such that is strictly increasing in

  is strictly decreasing in or vice versa. Then, has point of inflection at .
(ii)
 Let exist in , for some , except possibly at , such that for and for , or vice versa. Then, has a point of inflection at .
 
  Above theorems also give us a method of analyzing points of inflection.
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