Critical Points and Partial Derivatives

The video tutorial explains how to determine critical points of a function f(x, y) and classify them as relative maxima, minima, or saddle points. It involves finding first and second order partial derivatives, solving equations to find critical points, and applying the second partials test. The tutorial concludes with a graphical verification of the results using a 3D grapher.

What is the first step in determining the nature of critical points for a function of two variables?

• Use the mixed partial derivative test.

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

• Graph the function to visually identify critical points.

• Calculate the second order partial derivatives.

When finding the first order partial derivative with respect to x, what is treated as a constant?

• Both x and y

• Neither x nor y

• x

• y

What is the result of differentiating x^2 with respect to y?

• x

• 2y

• 0

• 2x

Which method is used to solve the system of equations to find critical points?

• Substitution

• Graphical

• Elimination

• Matrix

What is the critical point found in the system of equations for this function?

• (1, 1, 1)

• (0, 0, 0)

• (2, 2, 2)

• (1, 0, 0)

What is the formula used to calculate the value of D?

• D = f_xx * f_yy + (f_xy)^2

• D = f_xx * f_yy - (f_xy)^2

• D = f_xx - f_yy + (f_xy)^2

• D = f_xx + f_yy + f_xy

If D is greater than zero and the second order partial with respect to x is also greater than zero, what does the critical point represent?

• Saddle point

• Relative minimum

• Inconclusive

• Relative maximum

What is the value of D calculated for the critical point in this function?

• 9

• 3

• 7

• 5

What does the graphical representation confirm about the critical point?

• It is a saddle point.

• It is a relative maximum.

• It is a relative minimum.

• It is inconclusive.

In this context, what can the critical point also be classified as, given it is the lowest point on the surface?

• Absolute maximum

• Absolute minimum

• Saddle point

• Relative maximum