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Find and classify, using the second partial derivative test, the critical points of the function defined .$$f(x, y)=3 x^{2}-18 x+4 y^{2}-24 y+10$$

$$\mathrm{rm}(3,3,-53)$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

Partial Derivatives

Johns Hopkins University

Campbell University

University of Michigan - Ann Arbor

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.

04:14

02:34

Find and classify, using t…

01:29

02:03

02:48

02:56

01:51

01:48

02:01

03:06

02:18

01:23

01:57

01:22

01:59

01:39

01:04

for this problem we are asked to find and classify using the second partial derivative. Test the critical points of the function given. So the first thing that we need to do is find where the radiant of our function is going to equal zero. Now taking the partial derivative with respect to X. For the first element we get six x minus 18 And with respect to y gets eight y -24. Now solving for X and Y. Such that the radiant equals the zero vector. We'll get that X must equal three and Y must equal three. Having that. That suggests are critical point that we need to test is that .33. So continuing on to our test, we take the second partial derivative with respect to X, which gives us positive. six. Important note that that is greater than zero. Then we take the second partial derivative with respect to why? Which is going to give us eight. And we can see that the mixed partial derivative is going to be zero. So our D. Or at least that is what I call it. That we use for classification is going to be six times eight or 48 Which is greater than zero. And since we have that the second partial derivative with respect to X is greater than zero as well, that tells us that the .33 is a relative minimum

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