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I've drawn a crazy looking function here in yellow

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and what I want to think about is

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when this function takes on maximum values

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and minimum values.

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And for the sake of this video,

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we can assume that the graph of this function

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just keeps getting lower and lower and lower

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as x becomes more and more negative,

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and lower and lower and lower as x goes beyond

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the interval that I've depicted right over here.

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So what is the maximum value that this function takes on?

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Well, we can eyeball that. It looks like it's at this point,

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it looks like it's at that point right over there.

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So we would call this a global maximum.

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The function never takes on a value larger than this

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So we can say that we have a global maximum at the

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point x-nought. Because f(x)-nought is greater than

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equal to f(x) for any other x

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in the domain.

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And that's pretty obvious when you look at it like this.

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Now do we have a global minimum point

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the way that I've drawn it? Well no,

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this function can take on arbitrarily negative values

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it approaches negative infinity as x approaches

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negative infinity. It approaches negative infinity as

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x approaches positive infinity.

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So we have, let me write this down, we have

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no global minimum. Now let me ask you a question.

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Do we have local minima, or local maxima?

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When I say minima, it's just the plural of minimum.

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And maxima is just the plural of maximum.

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So do we have a local minima here, or a local minimum here?

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Well, a local minimum you can imagine means that

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the value of that funtion is lower than the points around it

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SO right over here it looks like we have a local minimum.

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We have a local minimum.

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And I'm not giving you a very rigorous definition here,

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but one way to think about it is, we can say that

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we have a local minimum point at x-one, is if we have

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a region around x-one where f(x)-one is less than f(x)

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for any x in this region right over here. And it's pretty

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easy to eyeball too. This is a low point for any

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values of f around it.

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Right over there.

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Now do we have any other local minima?

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Well it doesn't like we do.

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Now what about local maxima?

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Well, this one right over here, let me do it in purple.

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I don't want to get people confused, actually,

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let me do it in this color.

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This point right over here looks like a local

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maximum. Not lox, that would have to deal with salmon.

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Local maximum, right over there.

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So we can say at the point x-one, sorry, at the

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point x-two, we have a local maximum point at x-two,

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because f(x)-two is larger than f(x) for any x around

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a neighborhood around x-two.

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I"m not being very rigorous,

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but you can see it just by looking at it.

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So that's fair enough. We've identified all of the

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maxima and minima, often called the extrema,

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for this function.

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Now how can we identify those if we knew something

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about the derivative of the function?

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Well, let's look at the derivative at each of these points.

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So at this first point right over here, if I were to try

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to visualize the tangent line,

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let me do that in a better color than brown.

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If I were to try to visualize the tangent line, it

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would look something like that. So the slope here is a zero.

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So we would say that f-prime of x-nought is equal

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to zero.

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The slope of the tangent line at this point is zero.

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What about over here? Well, once again, the tangent line

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would look something like that.

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So once again we would say f-prime at x-one is equal

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to zero.

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What about over here? Well, here, the tangent line

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is actually not well-defined. We have a positive slope

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going into it and it immediately jumps into being

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a negative slope.

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So over here, f-prime of x-two is not defined.

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Let me just write "undefined."

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So we have an interesting, and once again, I'm not

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rigorously proving it to you. I just want you to get the

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intutiion here. We see that if we have some type of extrema,

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And we're not talking about when x is at an endpoint of

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an interval, just to be claerwhat I"m talking about.

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when I'm talking about x as an endpoint of an interval.

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We're saying that, let's say that the function is,

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let's say we have an interval from there,

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so let's say that a function starts right over there and then keeps going.

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This would be a maximum point, but it would be an endpoint.

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We're not talking about endpoints right now.

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We're talking about when we have points in between,

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or when our interval is infinite.

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So we're not talking about points like that, or points like

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this.

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We're talking about points in between.

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So if you have a point inside of an interval.

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It's going to be a mimum or maximum, and we see the

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intuition here.

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So, non-endpoint min or max at x is equal to a,

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so you know if you have a minimum or maximum point,

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at some point x is equal to a, and x is at the endpoint