Partial Derivatives and Critical Points

The video tutorial explains how to calculate critical points of a function and determine if they represent relative maxima, minima, or saddle points. It covers finding first and second-order partial derivatives, solving equations to find critical points, and applying the second partials test to identify the nature of these points. The tutorial concludes with a graphical verification of the results.

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

• Graphing the function

• Performing the second partials test

• Finding where the first order partial derivatives are zero or undefined

• Calculating the function's maximum value

When differentiating with respect to x, how is y treated?

• As zero

• As a function

• As a constant

• As a variable

What is the result of differentiating 6 divided by x with respect to x?

• -6x

• 6x

• 6/x^2

• -6/x^2

What is the purpose of finding second order partial derivatives?

• To solve the system of equations

• To find the function's domain

• To determine the function's maximum value

• To perform the second partials test

What is the mixed partial derivative with respect to x and then y?

• 30x to the fourth

• 12 divided by x cubed y

• 6 divided by x squared y squared

• 6x to the fifth

What is the equation obtained after clearing fractions in the first equation?

• 6x to the seventh y minus 6 equals zero

• 6y to the seventh x minus 6 equals zero

• 6y to the fifth minus 6 equals zero

• 6x to the fifth minus 6 equals zero

What are the critical points found in the system of equations?

• (1, 1) and (-1, -1)

• (2, 2) and (-2, -2)

• (0, 0)

• (3, 3) and (-3, -3)

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

• A relative minimum

• A relative maximum

• A saddle point

• An undefined point

What is the z-coordinate of the critical points found?

• 5

• 8

• 12

• 10

What does the graph of the surface verify about the critical points?

• They are relative minimums

• They are saddle points

• They are undefined

• They are relative maximums