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Find and classify, using the second partial derivative test, the critical points of the function defined .$$f(x, y)=x^{3}+y^{3}-9 x y$$

$$\operatorname{sp}(0,0,0), \operatorname{rm}(3,3,-27)$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

Partial Derivatives

Johns Hopkins University

Missouri State University

Baylor University

Boston College

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Find and classify, using t…

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for this problem we are asked to find and classify using the second partial derivative. Test the critical points of the function F of x, Y equals x cubed plus y cubed minus nine X. Y. To begin to find the critical points we want to solve for when the gradient of the function is going to equal zero. The first element of the gradient is going to be three, X squared minus nine. Why the second elements will be three, Y squared minus nine X. We want that to equal 00. So there are two possible solutions here. We have X equals zero, Y equals zero is one and we have X equals three. Y equals three. As the other. Now that we have are critical points. We can proceed with our test. So first step is to take our second partials Second partial X is going to be six X. Second partial Y Will be six Y. And the mixed partial is going to be the mix partial will be negative night having that. Then our d function that we use for evaluating or for testing rather it's going to be 36 X y minus 81. Now at the zero we can clearly see that we'll get negative 81 Less than zero, indicating that the zero is a saddle. Then evaluating at the .33, We get a final value of 243 Which is greater than zero. And we can see that the second partial X is going to be greater than zero as well, which indicates that we will have a actually already down below indicates that the .33 will be a relative minimum.

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