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

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

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

Â maximum value and the function attains a minimum value.

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

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

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

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

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

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

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

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

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

Â D is where the function achieves that maximum value.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â the wheels are important for the car, right?

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

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

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

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

Â function on the domain zero to infinity including zero.

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

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

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

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

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

Â values? Well no think about this example the

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

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

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

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

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

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

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

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

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

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

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

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

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

Â theorem by eliminating the condition at the function being continuous.

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

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

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

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

Â choose the maximum and minimum value. No.

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

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

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

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

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

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

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

Â So let's write I to represent some interval.

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

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

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

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

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

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

Â of I. We can use this terminology to rephrase

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

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

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

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

Â X. Then we look at the continuous function.

Â Remember to look at the image of some interval.

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

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

Â Well, the output can be anything. Alright?

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

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

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

Â closed? A much fancier way of stating the extreme

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

Â bounded interval is a closed bounded interval.

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

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

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

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

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

Â all the outputs whenever the input is in I.

Â Well this is going to be the closed interval.

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

Â achieves? And the biggest output is just going to be

Â whatever the maximum value is that this function achieves.

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

Â bounded interval is really getting at the same idea.

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

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

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

Â brackets here. The fact that these are closed interval

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

Â value is actually achieved. And the maximum value is actually

Â achieved. This way of talking is also useful for

Â something like the intermediate value theorum.

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

Â continuous image of an interval is an interval.

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

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

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

Â getting at. This is just a very succinct way of

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

Â there's something really special about continuity.

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

Â is really important. If someone just shows you some function

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

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

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

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

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

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

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

Â