Module 13 :  Maxima, Minima and Saddle Points, Constrained maxima and minima
Lecture 38 :  Second derivative test for local maxima / minima & saddle points [Section 38.1]
38.1.3 Theorem (Discriminant Test):
 

Assume that the first and second order partial derivatives of exist and are continuous at every point of Suppose

Then, we have the following:

(i)

The function has a

 
(ii)

The function has a

 

(iii)
The function has a
 

We assume his result, the interested reader may refer a book on advanced calculus book.

   
38.1.4 Example:
 

Let

Then,

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