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[music] Subject to some conditions functions are guaranteed to have maximum

and minimum values here is the extreme value thery it says that if a function F

is continuous under closed. Interval a, b, then the function attains a

maximum value and the function attains a minimum value.

If you don't like this, you can make it a little bit more precise.

Here's a slightly more precise statement of the extreme value theorem.

It says: "If a function f is continuous on the closed interval a, b," then this is

really just making clear what I mean by, "attains a maximum value and attains a

minimum value." All right? What I really mean is that there's some

point, c and d In this interval so that no matter what other value I pick in that

interval, f of c is less than or equal to that other value, and f of d is bigger

than that other value. So f of c is the minimum value.

C is where f achieves that minimum value. And f of d is that maximum value.

D is where the function achieves that maximum value.

The extreme value theorem promises us that these extreme values exist, but it doesn't

tell us how to find those extreme values. I mean, you could have a complicated

looking but continuous function on the closed interval between a and b.

And it might be hard to actually determine where the maximum value occurs, but the

Extreme Value Theorem tells you that the maximum value is achieved somewhere on

that closed interval. Nevertheless, even in that example, there

are extreme values, it's just hard to find them.

Now maybe that all seems obvious. One way to understand the extreme value

theorem is to try to break it. Try to mess with it someone and see if it

still works. This is sort of the car method to doing

science, right? If I'm studying cars and I want to know

what's important to making a car go. Well, I used to just take off the wheels

and then see if the car still works. Doesn't work so well, right?

The fact that the car doesn't work when I break the car's wheels off tells me that

the wheels are important for the car, right?

Same game with this theorem. I want to figure out what the important

parts of this theorem are so I'm going to try to break this theorem in various ways

and if the theorem doesn't after I break it, well that must have been an important

part of the theorem. What if, for instance, we were studying a

function on the domain zero to infinity including zero.

So, yeah, what if instead of closed interval between these two numbers a and

b, I'll have infinity be one of those numbers.

Then of course, then I have to make that an open parenthesis there.

So if a function's just continuous on the interval between zero and infinity

including zero, does that mean that it actually achieves its maximum minimum

values? Well no think about this example the

function F of X equals X define on that interval this function does not achieve a

maximum value. If you tell me that some number is the

maximum output of this vunction I'm just going to add one to wahtever you say and

that'll be a bigger to this function. This function does not acheive it's maxium

output at given input. What if we were studying a function on a

domain that was an open interval? Yeah, so what if instead of closed

interval, I just said open interval between these two numbers a and b?

Well then the function would not be guaranteed to have a maximum or minimum

value, then take a look at this. Here is an example of a continuous

function defined at an open interval minus pi over 2 and pi over 2 and this function

does not achieve a maximum or minimum value on this interval.

I can make tangent as positive or as negative as I'd like by choosing x close

enough to one of these end points. What if I try to mess up the extreme value

theorem by eliminating the condition at the function being continuous.

So yeah, what if I break the statement of the extreme value theorem by removing the

continuity equation because I'm not required the function be continuous

anymore. So, if it's just some function that's

defined on a close interval between a and b, does it guarantee the function that

choose the maximum and minimum value. No.

Here's an example. Here's a picture of a graph of a

discontinuous function and it's defined in the closed interval between a and b, but

here you can see that it's approaching some value but never achieving that value

and here it's approaching some value but never achieving that value.

So this function does not Actually attain its maximum and minimum value on this

interval. To get a better sense of what's going on,

here's a much fancier way to talk about the possible output values of a function.

So let's write I to represent some interval.

It could be the open interval between a and b, the close interval between a and b,

whatever, right? I is just a set of numbers.

What's f of a collection of numbers? What does that even mean?

Well this is the collection of all the output values if the input is in I.

So f of I is the set, or collection, of all the f of xs whenever x is some point

in the set I. And people sometimes call this the image

of I. We can use this terminology to rephrase

what we've been talking about. For example, the continuous image of a

bounded interval, a bounded interval like this interval between 2 numbers, need not

be bounded. Here's an explicit example that we've

already seen, take a look at this continuous function, Function, tangent of

X. Then we look at the continuous function.

Remember to look at the image of some interval.

Here is a interval. And what's the output of this function if

the input is between minus pi over 2, and pi over 2.

Well, the output can be anything. Alright?

So f of i is all real numbers. So even though I'm starting with things

that aren't very big. The output is as big or as small as I'd

like it to be. What it that bounded interval is also

closed? A much fancier way of stating the extreme

value theorem is to state this result. That the continuous image of a closed

bounded interval is a closed bounded interval.

You can really try this for yourself. You write down some crazy continues

function so who knows what this continues function might be some continues function

and some interval which I'll call I. >> Between two numbers A and B.

So it's bounded because A and B are numbers, right, and it's closed because

I'm using the square brackets. And then what I want to think is what is

all the outputs whenever the input is in I.

Well this is going to be the closed interval.

Right, what's the smallest value, what's the minimum value that the function

achieves? And the biggest output is just going to be

whatever the maximum value is that this function achieves.

So this statement that, continuous image of a closed, bounded interval is a closed,

bounded interval is really getting at the same idea.

It's the extreme value theorem, saying if you start with a continuous function on

some closed, bounded interval Then the ouptputs are going to be between.

Including the minimum and maximum value. And the fact that these are square

brackets here. The fact that these are closed interval

are crucial. Because I'm telling you that the minimum

value is actually achieved. And the maximum value is actually

achieved. This way of talking is also useful for

something like the intermediate value theorum.

So, yeah. You could say it's true that the

continuous image of an interval is an interval.

So if you take an interval and you pick your favorite continuous function, f of

that interval is just some other interval. That's the same as saying that over values

in-between are achieved and that's really what the intermediate value theory is

getting at. This is just a very succinct way of

stating the intermediate value theorem. The upshot here is that we're seeing

there's something really special about continuity.

So here again is the extreme value fare I mean what we've learned is that continuity

is really important. If someone just shows you some function

that's not continuous don't expect that you're going to be able to find any

extreme values they might not exist. It's also really important that this is

about closed interval between two numbers alright a closed and bound interval and

that's really important and if you're being asked to find maximum minimum values

not on a closed interval say you shouldn't expect that they necessary exist.

They might not be there. So this theorem's going to help guide you

towards the correct answer because it tells you what sort of answer to expect.