# Intermediate Value Theorem

How to solve intermediate value theorem problems: theorem, example, and its solution.

## Theorem

If *f*(*x*) is continuous in the interval [*a*, *b*]

and *f*(*a*) ≠ *f*(*b*),

then *c* exists in the interval (*a*, *b*)

that satisfies *f*(*c*) = *k*.

[*k* is between *f*(*a*) and *f*(*b*).]

If you look at the graph above,

you'll know the meaning of this theorem.

## Example

*f*(*x*) is a polynomial.

So *f*(*x*) is continuous in [1, 2].

(A polynomial function is continuous in real numbers.

So *f*(*x*) is continuous in [1, 2].)

*f*(1) = -2.

So *f*(1) is (-).

*f*(2) = 3.

So *f*(2) is (+).

So *f*(1) ≠ *f*(2).

Then, by the intermediate value theorem,*c* exists in (1, 2) that satisfies *f*(*c*) = 0.

[Of course 0 is between *f*(1), minus, and *f*(2), plus.]

So the zero of *f*(*x*) exists in (1, 2).

Let's see what you've done.*f*(*x*) is continuous in [1, 2].*f*(1) is (-).

And *f*(2) is (+).

So the zero of *f*(*x*), *c*, exists in (1, 2).