Critical Points and Partial Derivatives

The video tutorial explains how to find and classify critical points of a function using partial derivatives. It covers calculating first and second order partial derivatives, solving for critical numbers, and using the second partials test to determine if points are relative maxima, minima, or saddle points. The tutorial concludes with a graphical verification of the results.

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

• Performing the second partials test

• Finding the z-coordinates

• Locating the critical numbers

• Graphing the function

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

• f

• x

• z

• y

What is the derivative of 9x^2y with respect to x?

• 4x

• 18y

• 9x^2

• 18xy

How do you find the critical points of a function?

• By setting the second order partial derivatives to zero

• By solving the system of equations from the first order partial derivatives

• By graphing the function

• By finding the mixed partial derivatives

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

• 16/81

• -16/81

• 1

• 0

What is the result of substituting x = -4/9 and y = 2/9 into the function?

• 1

• 16/81

• -16/81

• 0

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

• A saddle point

• A relative maximum

• A relative minimum

• An inconclusive test

If D is negative, what type of critical point is indicated?

• Saddle point

• Relative minimum

• Inconclusive

• Relative maximum

What is the second order partial derivative with respect to y at (0, 0)?

• 4

• 0

• -8

• 18

What is the significance of the mixed partial derivative in the second partials test?

• It determines the z-coordinate

• It is used to calculate D

• It indicates a relative maximum

• It is not used in the test