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Find and classify, using the second partial derivative test, the critical points of the function defined .$$f(x, y)=x^{3}+6 x y+y^{2}$$

$$\operatorname{sp}(0,0,0), \mathrm{rm}(6,-18,-108)$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

Partial Derivatives

Harvey Mudd College

Baylor University

University of Nottingham

Idaho State University

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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02:18

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02:48

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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 six X Y plus Y squared. So our first step looks I need to actually be writing. Our first step is going to be to find X and Y. Such that the gradient of our function equals zero. Now the gradient or the zero vector the gradient is going to be three X squared plus six Y. In the first component and six X plus two Y in the second component we want that to equal 00. Now we're going to get out to solutions here. Either X equals zero and y equals zero or X equals six and y equals negative 18. So now that we have are critical points we can continue forward with doing our test. So we take the second partial derivative with respect to X. is going to give us six X. Take the second partial derivative with respect to why? Which is going to give us to And we take our mixed partial which is going to give us positive six than our D function. D X Y is going to equal 12 X minus 36. So we can see then that d. 00 It's going to be negative 36 Which is less than zero which tells us that the zero is a saddle point. Then evaluating it at the .6 -18. 6 -18. All that really matters is that our first term there is going to be 12 times six, which is going to be greater than 36 so that is going to be greater than zero, and our second partial derivative is going to be greater than zero as well, Which our second partial derivative test tells us. Then that's the .6 -18 will be a relative minimum.

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