Critical Points and Partial Derivatives

The video tutorial explains how to find and classify critical points of a function using first and second order partial derivatives. It covers the process of solving for critical points by setting partial derivatives to zero and using substitution. The second partials test is then used to determine if these points are relative minima, maxima, or saddle points. The tutorial concludes by calculating the D value for classification and finding the relative minimum values at the critical points.

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

• Evaluate the function at various points.

• Set the second order partial derivatives to zero.

• Calculate the mixed partial derivatives.

• Find where the first order partial derivatives are zero or undefined.

When finding the partial derivative of a function with respect to x, what should be done with y?

• Treat y as a constant.

• Ignore y completely.

• Treat y as a variable.

• Differentiate y with respect to x.

What method is used to solve the system of equations for critical points in this tutorial?

• Matrix method

• Substitution method

• Graphical method

• Elimination method

What is the result of factoring the equation 8y^3 + 288y = 0?

• y = 0, y = 6, y = -6

• y = 0, y = 12, y = -12

• y = 0, y = 3, y = -3

• y = 0, y = 9, y = -9

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

• 24y^2

• 2

• 0

• 24

What is the value of the mixed second order partial derivative with respect to x and then y?

• 2

• 0

• 24

• 24y^2

What does a negative D value indicate about a critical point?

• Inconclusive test

• Saddle point

• Relative maximum

• Relative minimum

If D is positive and the second order partial with respect to x is positive, what is the nature of the critical point?

• Inconclusive

• Saddle point

• Relative maximum

• Relative minimum

What are the relative minimum function values found in this tutorial?

• 72

• -2592

• 0

• 2592

Which of the following points is a relative minimum according to the tutorial?

• Both 72, -6 and -72, 6

• 0, 0

• 72, -6

• -72, 6