Module 4 : Local / Global Maximum / Minimum and  Curve Sketching
Lecture 11 : Convex / Concave Functions [Section 11.2]
11.2.6
 Theorem (First derivative test for convexity / concavity):
  Let be such that exists. Then the following holds:
  (i) is concave upward if and only if is increasing.
  (ii) is concave downward if and only if is decreasing.
 

(iii) If is strictly increasing, then is strictly concave upward.
  (iv) If is strictly decreasing, then is strictly concave upward.
   
  Proof:
 

We assume the proof. For details the reader may refer any book on Advanced Calculus/ Real Analysis.

 
   
   
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