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

Let . Then is convex if and only if for

                                                                       
 
 
11.2.4 Theorem:
 

be a function which is concave upward/downward. Then, the following holds:

(i)
is continuous.
(ii)

For every has both the left derivative, and the right derivative

 

Further, for

 
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