[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.