# Rolle's Theorem > In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero. The theorem is named after Michel Rolle. > > [Wikipedia](https://en.wikipedia.org/wiki/Rolle's%20theorem) In other words, if $f(a) = f(b)$ there must exist at least one point $c$ between $a$ and $b$ such that $f'(c) = 0$. Rolle's theorem is applicable to all [[Smooth Functions]]. ## Proof Several cases exist. The function could be linear, in which case its every point has derivative equal to 0. The two points $a$ and $b$ could be the function's absolute maximum, or its absolute minimum, or neither. In all three cases [[The Extreme Value Theorem]] can be applied to ascertain that the at least one point exists where the derivative is 0 1. The two points are the function's absolute maximum. All other points have values that a less than $f(a)$. Per EVT, the candidates for the local minimum are $a$, $b$, and points where the derivative is 0. There must be some values that are less than the absolute maximum, so there must be at least one point where the derivative is 0. 2. The two points are the function's absolute minimum. Same reasoning can be applied as in case 1 but flipped. 3. The two points are neither the function's absolute maximum, nor its absolute minimum. Because the function is smooth, and because it is not linear, there must be some values between $a$ and $b$ that are bigger than $f(a)$. Again per EVT, there must be some points where the derivative is 0 (since we excluded $a$ and $b$ as local maximums).