Review of a function of two variables: Maxima and Minima

 Maxima and Minima Problems

In this post we review the following notions:

Critical points in a function of three variables

Local extremum and the saddle point in a function of two variables

Absolute maxima and minima

Determination of the local extremum and the saddle point of a function of two variables

 Local and global extremum in a function of two variables

The extremum of a function is either a minimum or a maximum. The purpose of determining a critical point is to determine relative maxima or minima of a function. In a function of a single variable, we determine an interval around the critical point where the value of the function for that critical point is greater or less than all the values of the function in the chosen interval. For a function of two variables we do the same consideration with the exception that the interval is a disk.

Definition 

Saddle point. Definition

Second Derivative Test. Theorem

Example

Find the critical point of the critical points of the following function. Use the second derivative test to find the local extrema.

Solution

Practice