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For $f(x, y)=3 x^{2}-18 x+4 y^{2}-24 y+10,$ find the point(s) at which $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$.

$(3,3)$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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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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So if we want to find the points that make both of our partial derivatives equal to zero, we first just have to find our partial derivatives seven micro zero and then solve for the values of X and Y and then kind of take the intersection since we want both of them to be zero. So over here I'll do del by Dell experts Final partial with respect to X B f sub X is equal to so remember all these wise here, we assume, are constants. Eso if I square constant, multiply it by a constant still constant. And so the derivatives of these are all going to be zero. And then those first two we just use power rule would be six X minus 18 on Now we're going to go ahead set the secret zero solve for X so that would imply X is equal to three on DNA. Now, actually, let me scoop this down a little bit. First this up so we know it X is equal to three on. It doesn't really matter what. Why is it this point? We will have where are partial derivative with respect? Ax is zero. So it's just kind of break that out. So f sub x of three y. That should be zero right now. If we go ahead and do the partial of this with respect, why eso we get eps of Why is equal to now we're going to treat thes access. Here is if there's Constance, just like we put this 10. And so again, For that reason, these two are just going to be zero. And then to take those, we would use power rule s O B eight y minus 24. Um, then we go ahead, set this equal to zero, which is going to imply, um, so we'd add 24 divide by eight. So why is even 23 which is going to say that or any value X we can plug in the value of why being three and then our partial is going to be zero. And so now the only intersection that this has because essentially, this first solution set is just all points three y and then over here, this is all of the points x three. And if we go ahead and intersect those two groups, the only point that will ensure that both of these is going to be zero is the 00.3 three and so that this would be our solution.

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