# Rolle's theorem

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## Statement

Suppose $f$ is a function defined on a closed interval $[a,b]$ (with $a < b$) satisfying the following three conditions:

1. $f$ is a continuous function on the closed interval $[a,b]$. In particular, $f$ is (two-sided) continuous at every point in the open interval $(a,b)$, right continuous at $a$, and left continuous at $b$.
2. $f$ is differentiable on the open interval $(a,b)$, i.e., the derivative of $f$ exists at all points in the open interval $(a,b)$.
3. $f(a) = f(b) = 0$.

Then, there exists $c$ in the open interval $(a,b)$ such that $f'(c) = 0$.

## Proof

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 If $f$ is zero on all of $[a,b]$, then $f'(c) = 0$ for any choice of $c\in (a,b)$ obvious
2 $f$ must attain both its maximum and its minimum values on $[a,b]$ Fact (1) $f$ is continuous on $[a,b]$ Given-fact combination direct
3 If $f$ is not zero on all of $[a,b]$, either its absolute maximum value on $[a,b]$ is positive and attained at a point in the open interval $(a,b)$ or its absolute minimum value on $[a,b]$ is negative and attained at a point in the open interval $(a,b)$ (or possibly both). $f(a) = f(b) = 0$ Step (2) [SHOW MORE]
4 If $c$ is a point in $(a,b)$ at which $f$ attains its maximum value or its minimum value, then $\! f'(c) = 0$. Fact (2) $f$ is differentiable on $(a,b)$ [SHOW MORE]
5 There is a point $c\in (a,b)$ at which $\! f'(c) = 0$. Steps (1), (3), (4) [SHOW MORE]