Critical Points and Local Extrema

This video tutorial explains how to identify critical points, saddle points, and local extrema in multivariable functions. It outlines the steps to find these points using first and second partial derivatives. The video includes two example problems that demonstrate the process of calculating and analyzing these points to determine if they are local minima, maxima, or saddle points.

What is the first step in identifying critical points in a multivariable function?

• Calculate the second partial derivatives

• Evaluate the function at a point

• Find the mixed partial derivative

• Set the first partial derivatives to zero

What does a positive value of D indicate when analyzing a point?

• The point is a local maximum

• The point is a saddle point

• The point is a local minimum or maximum

• The point is neither a local min nor max

In the first example, what is the value of the function at the local minimum point (2, 2)?

• 24

• 30

• 12

• 8

What is the mixed partial derivative f_xy in the second example problem?

• 8

• 24

• 0

• -8

In the second example, what is the value of D at the point (1, 1)?

• 576

• 512

• 64

• 0

What is the significance of a negative D value?

• The point is a local minimum

• The point is neither a local min nor max

• The point is a local maximum

• The point is a saddle point

What is the function value at the local minimum point (1, 1) in the second example?

• 24

• 12

• 16

• 8

How many local minima are identified in the second example problem?

• None

• Two

• One

• Three

What is the function value at the local minimum point (-1, -1) in the second example?

• 8

• 12

• 24

• 16

What is the saddle point identified in the second example problem?

• (1, 1)

• (2, 2)

• (0, 0)

• (-1, -1)