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Find and classify, using the second partial derivative test, the critical points of the function defined .$$f(x, y)=2 x y^{2}+2 x^{2} y-8 x+4$$

$$\operatorname{sp}(0,\pm 2,4)$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

Partial Derivatives

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University of Michigan - Ann Arbor

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Lectures

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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 shown below. Now our first step is going to be to try to find X and Y such that the gradient equals zero. That will be our critical points. So the first element by taking the partial derivative with respect to X is going to be two, Y squared plus four, X y minus eight. And the first partial with respect to y is going to be two, X squared plus four X. Y. That's eight. And there we go. And we want that to equal the zero vector. So we have a system of two equations with two unknowns and we get out actually to solutions X equals zero and y equals plus or minus two. So now that we have our critical points, we continue with our test. So what we need to do is take the second partial derivative with respect to X. Which is going to give us for Y. The 2nd partial derivative with respect to why? Which is going to give us for X. And the mixed partial derivative D X Y. Which is going to give us four X plus four Y. So our d function is going to be um It's going to be here 16 XY four x plus four y. All square now, one moment here Evaluating at the .02. Actually since we are actually no, I'm going to take that back evaluating at the .02. We get a value of negative 64 less than zero indicating that 02 is a saddle. All right saddle point. And evaluating at the 0.0.0 negative two, we actually get out negative 64 again, Which implies that 0 -2 is going to be a saddle as well.

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