of the continuous function f.

All right, fair enough.

Can we use the intermediate value theorem

to say that the equation f of x equal is equal to zero

has a solution where four

is less than or equal to x is less than or equal to six?

If so, write a justification.

So, pause this video and see if you can think about this

on your own before we do it together.

Okay, well let's just visualize what's going on

and visually think about the intermediate value theorem.

So, if that's my y-axis there

and then let's say that this is my x-axis

right over here.

We've been given some points over here.

We know when x is equal to zero, f of x is equal to zero.

Let me draw those.

So, we have that point.

When x is equal to two,

y or f of x, y equals f of x

is gonna be equal to a negative two.

So, we have a negative two right over there.

When x is equal to four, so, three, four,

f of x is equal to three.

One, two, three.

I'm doing it on a slightly different scale

so that I can show everything.

And when x is equal to six, so, five, six,

f of x is equal to seven.

Three, four,

five, six, seven.

So, right over here.

Now, they also tell us that our function is continuous.

So, one intuitive way of thinking about continuity

is I can connect all of these dots

without lifting my pencil.

So, the function might look,

I'm just gonna make up some stuff,

it might look something,

anything like what I just drew just now.

And it could have even wilder fluctuations

but that is what my f looks like.

Now, the intermediate value theorem

says hey, pick a closed interval.

And here, we're picking the closed interval

from four to six, so let me look at that.

So, this is one, two, three, four here,

this is six here,

so we're gonna look at this closed interval.

And the intermediate value theorem tells us

that look, if we're continuous over that closed interval,

our function f is gonna take on every value

between f of four, which in this case,

so, this is f of four, is equal to three,

and f of six, which is equal to seven.

f of six,

which is equal to seven.

And so, if someone said hey, is there gonna be a solution

to f of x is equal to, say, five over this interval?

Yes.

Over this interval, for some x,

you're going to have f of x being equal to five.

But they're not asking us for an f of x

equaling something between these two values.

They're asking us for an f of x equaling zero.

Zero isn't between f of four and f of six,

and so we cannot use the intermediate value theorem here.

And so, if we wanted to write it out,

we could say f is continuous

but zero is not

between f of four

and f of six.

So, the intermediate value theorem does not apply.

All right, let's do the second one.

So here they say, can we use the intermediate value theorem

to say that there is a value c such that f of c equals zero

and two is less than or equal to c

is less than or equal to four?

If so, write a justification.

We are given that f is continuous, so let write that down.

We are given

that f is continuous,

and if you wanna be over that interval,

but they're telling us it's continuous in general.

And then we can just look at what is the value

of the function at these end points?

Our interval goes from two to four,

so we're talking about this closed interval right over here.

We know that f of two

is going to be equal to negative two.

We see it on that table.

And what's f of four?

f of four is equal to three.

So, zero

is between

f of two and f of four.

And you can see it visually here.

There's no way to draw between this point and that point

without picking up your pen, without crossing the x-axis,

without having to point

where your function is equal to zero.

And so, we can say

according to the intermediate value theorem,

there is

a value c

such that

f of c is equal to zero

and two is less than or equal to c

is less than or equal to four.

So, all we're saying is hey, there must be a value c,

and the way I drew it here, that c value is right over

where c is between two and four,

where f of c is equal to zero.

And this seems all mathy

and a little bit confusing sometimes

but it's saying something very intuitive.

If I had to go from this point to that point

without picking up my pen, I am going to at least cross

every value between f of two and f of four at least once.