Critical Points and Partial Derivatives

The video tutorial explains how to calculate critical points of a function and classify them as relative maximum, relative minimum, or saddle points. It covers finding first and second order partial derivatives, solving a system of equations to locate critical points, and using the second partials test for classification. The tutorial concludes with a summary of the results, identifying relative minimums and a saddle point.

What is the first step in finding critical points of a function?

• Performing the second partials test

• Graphing the function

• Finding where the first order partial derivatives are zero or do not exist

• Evaluating the function at various points

When differentiating with respect to x, what is treated as a constant?

• y

• Neither x nor y

• x

• Both x and y

What is the second order partial derivative with respect to x?

• 8y cubed

• 32xy

• 12x squared

• 4x cubed

What substitution is made to solve the system of equations for critical points?

• x for y squared

• y squared for negative four x

• x cubed for y

• y for x squared

What are the x-values of the critical points found?

• 0, 4, -4

• 3, -3, 0

• 1, -1, 2

• 2, -2, 3

What is the z-coordinate of the critical point at (0, 0)?

• 16

• -256

• 0

• 256

What is the function value at the critical point (-4, 4)?

• -256

• 0

• 16

• 256

What does a D value of zero indicate in the second partials test?

• Inconclusive test

• Relative maximum

• Saddle point

• Relative minimum

What is the classification of the critical point at (0, 0, 0)?

• Saddle point

• Relative minimum

• Inconclusive

• Relative maximum

What is the relative minimum value of the function?

• 0

• 16

• -256

• 256