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For $f(x, y)=2 x^{2}-8 x+3 y^{2}+12 y+2,$ find the point(s) at which $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$.

$(2,-2)$

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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02:02

00:51

So if we want to find where he's partial, derivatives will be equal to zero act or we'll both of them will be. First we have to actually find the partial. So let's go ahead and do that. So here will do, Del by Dell X to get our personal with respect to X. Um, so remember, we're going to treat all of the wise is constant. So this, like, Why square this? Why and then essentially this to here are all going to be constant. So if we take the derivative of a constant directly, that's just going to be zero. Um, it's actually let me not right over this bill. This would be 000 and then here this is going to just be powerful. So would be four x minus eight, Um, and then just plus zero plus zero plus zero. Or, in other words, we just have where are partial derivative is when the four x minus eight. So now let's go ahead and set this equal to zero. And so then that implies will be at eight. Divide by four X is going to be too, so we at least know if we plug to into our partial derivative. With respect X, we would get where it is zero. And then why could be pretty much anything. Why all real numbers? But right now and now, we could go ahead and do the same thing. But this time, instead of taking the partial derivative with respect to X, we take partial derivative with respect to why So now we're going to treat thes excess Here is constant, just like we with that too. And actually, if we take the derivative, that's we get f sub y. So it's zero plus zero and then we use powerful for that s 06 y and then plus 12 plus zero. And so if we said this equals zero, we would subtract 12, divide by six. So that would imply why is it good to negative two? And then in this case, X could be any real numbers, since we don't have any restrictions on that. Mhm. So if we plug in So I actually kind of let me write out what we have here so f sub x of two. Why is equal to zero, and then we also have that f of just any value for X negative, too. Sub Y is equal to zero. And now we would just need to take the intersection of these two sets, which in this case, the only value that would make both of these zero is just going to be the point too negative, too. So this is going to be the Onley point that would make both of these equal to zero.

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