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Find the critical points.$$f(x, y, z)=2 x^{2}-16 x+3 y^{2}-24 y+4 z^{2}-40 z+25$$

$$(4,4,5,-155)$$

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.

04:14

00:59

Find the critical points.<…

01:20

For this problem, we are asked to find the critical points of that really long function shown below. So the critical points will be found where the gradient of said very long function equals the three dimensional zero factor. So the partial derivative of the function with respect to X. It's going to be four x minus 16, giving us the first element of the gradient with respect to why it's going to be six y minus 20 for giving us the second element. And with respect to Z, it's going to be a z minus 40 giving us the third element. Now, if we try to solve for where this equals the 33 dimensional zero vector, Then we'll find that x equals four. Why equals four and z equals five, giving us our critical point 445.

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