# Bolzano's Theorem

A paticular case of the the **intermediate value theorem** is the **Bolzano's theorem**:

**Suppose that f(x) is a continuous function on a closed interval [a, b] and takes the values of the opposite sign at the extremes, and there is at least one c (a, b) such that f(c) = 0.**

The **Bolzano's theorem **does not indicate the value or values of * c*, it only confirms their existance.

#### Example

Verify that the equation x³ + x − 1 = 0 has at least one real solution in the interval [0,1].

We consider the function f(x) = x³ + x − 1, which is continuous on [0,1] because it is polynomial. We study the sign in the extremes of the interval:

First, consider the function f(x) = x³ + x − 1, which is continuous in [0,1] because it is polynomial. Then, study the sign in the extremes of the interval:

f(0) = −1 < 0

f(1) = 1 > 0

As the signs are different, **Bolzano's theorem** can be applied which determines that there is a **c**** (0. 1)** such that **f(c) = 0**. This process demonstrates that there is a solution in this interval.