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## Extreme value theorem, global versus local extrema, and critical points

Current time:0:00Total duration:7:53

# Critical points introduction

## Video transcript

I've drawn a crazy looking
function here in yellow. And what I want
to think about is when this function takes
on the maximum values and minimum values. And for the sake
of this video, we can assume that the
graph of this function just keeps getting lower
and lower and lower as x becomes more and more
negative, and lower and lower and lower as x goes
beyond the interval that I've depicted
right over here. So what is the maximum value
that this function takes on? Well we can eyeball that. It looks like it's at that
point right over there. So we would call this
a global maximum. the? Function never takes on
a value larger than this. So we could say that we have a
global maximum at the point x0. Because f of of x0 is
greater than, or equal to, f of x, for any other
x in the domain. And that's pretty obvious,
when you look at it like this. Now do we have a
global minimum point, the way that I've drawn it? Well, no. This function can take an
arbitrarily negative values. It approaches
negative infinity as x approaches negative infinity. It approaches
negative infinity as x approaches positive infinity. So we have-- let me
write this down-- we have no global minimum. Now let me ask you a question. Do we have local
minima or local maxima? When I say minima, it's
just the plural of minimum. And maxima is just
the plural of maximum. So do we have a local minima
here, or local minimum here? Well, a local minimum,
you could imagine means that that value of the
function at that point is lower than the
points around it. So right over here, it looks
like we have a local minimum. And I'm not giving a very
rigorous definition here. But one way to
think about it is, we can say that we have a
local minimum point at x1, as if we have a region
around x1, where f of x1 is less than an f of x for any x
in this region right over here. And it's pretty easy
to eyeball, too. This is a low point for any
of the values of f around it, right over there. Now do we have any
other local minima? Well it doesn't look like we do. Now what about local maxima? Well this one right over
here-- let me do it in purple, I don't want to get
people confused, actually let me do it in this color--
this point right over here looks like a local maximum. Not lox, that would have
to deal with salmon. Local maximum, right over there. So we could say at the point
x1, or sorry, at the point x2, we have a local
maximum point at x2. Because f of x2 is larger
than f of x for any x around a
neighborhood around x2. I'm not being very rigorous. But you can see it
just by looking at it. So that's fair enough. We've identified all of the
maxima and minima, often called the extrema, for this function. Now how can we identify
those, if we knew something about the derivative
of the function? Well, let's look
at the derivative at each of these points. So at this first
point, right over here, if I were to try to
visualize the tangent line-- let me do that in a
better color than brown. If I were to try to
visualize the tangent line, it would look
something like that. So the slope here is 0. So we would say that f
prime of x0 is equal to 0. The slope of the tangent
line at this point is 0. What about over here? Well, once again,
the tangent line would look something like that. So once again, we would say
f prime at x1 is equal to 0. What about over here? Well, here the tangent line
is actually not well defined. We have a positive
slope going into it, and then it immediately jumps
to being a negative slope. So over here, f prime
of x2 is not defined. Let me just write undefined. So we have an interesting-- and
once again, I'm not rigorously proving it to you, I just want
you to get the intuition here. We see that if we have
some type of an extrema-- and we're not
talking about when x is at an endpoint
of an interval, just to be clear what I'm
talking about when I'm talking about x as an endpoint
of an interval. We're saying, let's
say that the function is where you have an
interval from there. So let's say a function starts
right over there, and then keeps going. This would be a maximum point,
but it would be an end point. We're not talking about
endpoints right now. We're talking about when
we have points in between, or when our interval
is infinite. So we're not talking
about points like that, or points like this. We're talking about
the points in between. So if you have a point
inside of an interval, it's going to be a
minimum or maximum. And we see the intuition here. If you have-- so non-endpoint
min or max at, let's say, x is equal to a. So if you know that you have
a minimum or a maximum point, at some point x is
equal to a, and x isn't the endpoint
of some interval, this tells you
something interesting. Or at least we
have the intuition. We see that the derivative
at x is equal to a is going to be equal to 0. Or the derivative at x is equal
to a is going to be undefined. And we see that in
each of these cases. Derivative is 0, derivative
is 0, derivative is undefined. And we have a word for these
points where the derivative is either 0, or the
derivative is undefined. We called them critical points. So for the sake
of this function, the critical points are,
we could include x sub 0, we could include x sub 1. At x sub 0 and x sub
1, the derivative is 0. And x sub 2, where the
function is undefined. Now, so if we have a
non-endpoint minimum or maximum point, then it's going
to be a critical point. But can we say it
the other way around? If we find a critical point,
where the derivative is 0, or the derivative is
undefined, is that going to be a maximum
or minimum point? And to think about that, let's
imagine this point right over here. So let's call this x sub 3. If we look at the tangent
line right over here, if we look at the
slope right over here, it looks like f prime of
x sub 3 is equal to 0. So based on our definition
of critical point, x sub 3 would also
be a critical point. But it does not appear to be
a minimum or a maximum point. So a minimum or maximum
point that's not an endpoint, it's definitely going
to be a critical point. But being a critical
point by itself does not mean you're at a
minimum or maximum point. So just to be clear
that all of these points were at a minimum
or maximum point. This were at a critical
point, all of these are critical points. But this is not a
minimum or maximum point. In the next video, we'll
start to think about how you can differentiate,
or how you can tell, whether you have a minimum or
maximum at a critical point.