# Using the Bolzano's theorem to prove that exists only one solution in the interval.

Considering the Bolzano's theorem: Let $f$ be continuous at each point of a closed interval $[a, b]$ and assume that $f(a)$ and $f(b)$ have opposite signs. Then there is at least one $c$ in the open interval $(a, b)$ such that $f(c) = 0$.

How can I use the Bolzano's theorem to prove that exists only one solution?

For example:

$f(x) = 25 + 60 e^{-0.35x}$

how can I prove that exists only one solution for $f(x)=40$ in the interval $x \in ]3,4[$ ?

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Do you mean using a combination of Bolzano's theorem and Rolle's theorem? –  mfl Jun 16 '14 at 13:33
Only with Bolzano's theorem. I found the solution in the first answer: i need to prove that f is strictly monotonically decreasing. –  Jorge Jun 16 '14 at 13:37

If the function $f$ is in addition strictly monotonically increasing (or decreasing as your example), then there is exactly one solution.