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

               
 
Proof:
 


We assume the proof. Interested reader may refer a book on advanced calculus or Real Analysis.

We shall show next how convexity/ concavity of is related to various properties of or .

   
   
 
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