Justification with the intermediate value theorem: equation | AP Calculus AB | Khan Academy

Let g of x equal one over x. Can we use the intermediate value theorem to say that there is a value c such that g of c is equal to zero and negative one is less than or equal to c is less than or equal to one?

If so, write a justification.

So in order to even use the intermediate value theorem, you have to be continuous over the interval that you care about, and this interval that we care about is from x equals negative one to one.

And one over x is not continuous over that interval. It is not defined when x is equal to zero.

And so we could say no because g of x is not defined. Not defined, or I could say let me just say not continuous.

It's also not defined at every point of the interval, but let's say not continuous over the closed interval from negative one to one.

And we could even put parentheses: not defined at x is equal to zero.

All right, now let's ask the second question. Can we use the intermediate value theorem to say that the equation g of x is equal to three-fourths has a solution where one is less than or equal to x is less than or equal to two?

If so, write a justification.

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 on the closed interval from one to two.

And if you wanted to put more justification there, you could say g is defined for all real numbers such that x does not equal zero.

I can write g of x is defined for all real numbers such that x does not equal to zero.

And you could say rational functions like one over x are continuous at all points in their domains.

At all points in their domain, that's really establishing that g of x is continuous on that interval.

And then we want to see what values does g take on at the endpoint, or actually these are the endpoints that we're looking at right over here.

g of one is going to be equal to one over one, which is one, and g of two is going to be one over two, which is equal to one over two.

So three-fourths is between g of one and g of two.

So by the intermediate value theorem, there must be an x that is in the interval from one to two such that g of x is equal to three-fourths.

And so yes, we can use the intermediate value theorem to say that the equation g of x is equal to three-fourths has a solution, and we are done.