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Linearize the given function.Show that the critical point for $f(x, y)=\frac{a}{x}+\frac{b}{y}+c x y$ occurs when $x=\left(\frac{a^{2}}{b c}\right)^{1 / 3}$ and $y=\frac{b}{a} x$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

Partial Derivatives

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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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Let $f(x, y)=y^{2} x-y x^{…

02:32

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03:20

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02:06

The critical points of the…

01:33

Find the critical points o…

02:22

Consider the functions $f(…

01:27

02:09

Okay, so consider the function F of X Y is equal to y squared X minus Y x squared plus X y. Okay, so let's find the partial with respect to X. So the parcel with respect to X of y squared X is just y squared the parcel with respect the X Why X squared is minus two x y the portal with respect to X. Here is just what? Okay, so noticed that I could pull away out everywhere. And I have why minus two x bliss. Okay, so then we find the partial respect. Why go back to the original function? So I have to Why? Ex minus, uh, X squared and then plus X so he reckon factor at X and I have to. Why? Minus X plus one. Okay, so we want to find the critical points s So we have a couple cases. First case is ah X and y are both equal dizzy round. So that's just see Rosie. Okay, The second case is ah, that X is equal to zero. And why is equal to if I plug this into why minus two X plus one is equal to zero or why plus one is equal to zero. Or why is equal a negative one. Okay, so we have zero negative one case three we have. Why is equal to zero now? Why is equal to zero then? That means that too. Why minus X plus one is equal to zero or X is equal to one. Okay, so this is the 0.10 So these are the critical numbers that we have thus far. Okay, so for the last one, it the last critical number we're gonna find by solving these two said equal to zero solving that system. Okay. So we can solve the first equation for why? Maybe so. Why is equal to two X minus one if I move everything to the other side? So then I have two times two X minus one minus X plus one is equal to zero or four x minus two minus X plus one is equal to zero. So three X um minus one is equal to zero or X is equal to 1/3. Okay, I'm so why is equal to 2/3 minus one, which is negative 1/3. So we have a critical point at 1/3 negative. One thing. Okay, so next we need to find the second partials. So we need to find f x X. We need to find f. Why? Why? And we need to find a excellent Okay, So if I take the derivative of FX with respect to X, I just have a minus two. Why? If we take the derivative of the Y one with respect, why we just have positive to X, And if I take the parcel of acts with respect to why, then I have to. Why? Minus two X plus one. Okay, so our function d hope is classified. The critical numbers is minus four x Y minus to why minus two X plus one squared in case it. Let's classify, uh oh. Four of these critical points. Okay, So if I pull again zero zero Republican 00 I get, um, minus one. Okay, so we know that that's a saddle point. If I plug in, um, 10 I end up with 00 I have minus, uh, negative two plus one. Oh, squared. Okay, so this is negative to the societal point. If I plug in zero negative one, then I have, um we're putting an X is equal to zero. So we have minus. We're playing in the negative too. Plus one squared, which is also listen. Zero. So that's a cell. Okay, so if we plug in, um, are critical point, uh, 1/3. Negative. 1/3. Okay, so when we plug this in, uh, it comes out to 1/3 which is positive. Um, so then we need to check f x x 1/3 negative 1/3 and that comes out to 2/3 which is positive. So we know that there is a local minimum at 1/3 negative one.

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