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Verify the critical point found in Example 8 is a relative minimum.

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

02:03

Determine whether the func…

01:58

02:50

02:04

00:39

Locate the critical points…

for this problem, we have been given the graph of a function. We have been asked to find the critical numbers Associated with this graph and determine if there are relative or absolute minimums or maximums at these points. Now, let's review what a critical number is. But critical numbers happen where the derivative of my function is zero or it's undefined. X. Think about what those mean. Undefined often means that we have some kind of point or cusp or it comes up to a point or down to a point. We have that sharp curve or a spot where maybe there's a disconnect, there would be undefined at that point as well if my derivative is zero, that's often a minimum or a maximum point, because that means my slope is zero. So we're looking for places where the slope is zero where it's undefined. So let's take a look at this problem here. This looks like a parabola. And if I was going to approximate the point on the problem, I would say that's X equals two and y equals four. So that would be that point where the slope levels off. I have a slope of zero. That's that first case on the function. So what kind of the point is this? Well, that is the highest value of the function on my graph here. So this is an absolute maximum. Now this graph does not have any absolute minimums because it's an open graph. If you look at as this graph comes down on either side, you can see those open end points at the bottom right where it hits the X axis. So the value of my function is going to get closer and closer and closer two, the X axis, but it's never going to hit there. So we this would not have a minimum value, It just has the absolute maximum.

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