Rolle's theorem

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 $f^{\prime }(c)=0$.

Related facts

Applications

• Lagrange mean value theorem
• Bound relating number of zeros of function and number of zeros of its derivative

Facts used

1. Extreme value theorem
2. Point of local extremum implies critical point