June 26, 2019
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.
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.