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Linearize the given function.$$f(x, y)=x^{4}+y^{4}+16 x y \text { near }(1,-1)$$

$$f(x, y)=10-12 x+12 y$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

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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01:38

Linearize the given functi…

03:33

Find a function $y=f(x)$ f…

05:53

Determine the function $f$…

01:27

Find $f^{\prime}(x)$ if $f…

01:02

Evaluate the given functio…

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01:34

Find $f$$$f^{\prime \p…

00:24

Find $f^{\prime}(x)$.$…

for this problem, we are asked to linearize the function F x Y equals X power four plus. What's the power four plus 16 X. Y near the 160.1 negative one. Now below, I have the general equation for linear ization. So actually, one moment here. All right. So let's first evaluate our function at the given point. So f of one negative one is going to come out to negative 14 than our partial derivative of the function with respect to X. At a generic point X Y Is going to become four x cubed plus 16 y. Partial derivative with respect to Y at X Y is going to be 16 X plus four Y cube. So putting everything Oh yeah, putting everything together into our linear ization equation. We will have -14. Then uh fx or the partial uh personal with respect to X evaluated at one negative one is going to be negative 12. So we have negative 14 minus 12 times X minus one. Then the partial with respect to why evaluated at 1 -1 is going to be positive 12. So we have plus 12 times Y plus one giving us our linear ization.

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