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Find all the critical points for each function.$f(x, y)=4 x y-x^{4}-y^{4}+2$

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$(0, 0) (1, 1)$ and $(-1, -1)$

Calculus 3

Chapter 13

Directional Derivatives, Gradients, and Extrema

Section 3

Extrema of Functions of Two Variables

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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Find all the critical poin…

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Find all critical points o…

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Find and classify the crit…

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in this problem we will cover extreme of two variable functions. So to find the critical points of this function F. We have to find the partial derivatives with respect to X and Y. And sent them equal to zero. So we begin with our partial derivative with respect to X. And that's just taking the derivative of the function holding. Why fixed? So we get four y minus for execute. And we set this equal 20. Now taking the partial derivative with respect to why? So they can lead derivative holding expects now We get four x minus for why? Cute. And we set this equal 20. And from these two let's see equations, we get different options for X and Y. So I see off that that we can get X equals one While why equals one. So we know that one critical point is going to be 11. We know that when X equals zero we can get Y equals zero. So that means we have another critical point at 00 And last we get that one x Is -1. We can have why equals negative one. And that means that our last critical point Is going to be -1 -1.

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