Name: Justification with the intermediate value theorem: equation | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T14:15:15.000Z
Duration: 3 min 46 s
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- [Instructor] Let g of x equal one over x.

Can we use the intermediate value theorem

and negative one is less than or equal to c,

So in order to even use the intermediate value theorem,

you have to be continuous over the interval

And this interval that we care about is from x equals

And, one over x is not continuous over that interval,

it is not defined when x is equal to zero.

g of x not defined, or I could say not continuous.

It's also not defined on every point of the interval,

but let's say not continuous over the closed interval

And we could even put in parentheses not defined,

All right, now let's start asking the second question.

Can we use the intermediate value theorem to say

has a solution where 1 is less than or equal to x,

All right so first let's look at the interval.

If we're thinking about the interval from one to two,

well, yeah, our function is going to be continuous

over that interval, so we could say g of x is continuous

And if you wanted to put more justification there,

you could say g defined for all real numbers,

I could write g of x defined for all real numbers

and you could say rational functions like one over x,

are continuous at all points in their domains.

That's going really establishing that g of x

And then we wanna see what values does g take over,

these are the end points we're looking at right over here.

G of one is going to be equal to one over one is one,

and g of two is going to be one over two.

So, 3/4 is between g of one and g of two,

there must be an x that is in the interval

from where it's talking about the interval from one to two,

And so, yes, we can use the intermediate value theorem

to say that the equation g of x is equal to 3/4