Intermediate Value Theorem

Intermediate Value Theorem

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

Theorem

If f(x) is continuous in [a, b] and f(a) is not equal to f(b), then c exists in (a, b) that satisfies k = f(c). [k is between f(a) and f(b).]

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

Show that the zero of f(x) exists in (1, 2). f(x) = x^3 - 2x - 1

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).