The theorem is named after the seventeenth century French mathematician Michel Rolle (16521719).
Theorem
Let f be a continuous function defined on [a, b] and differentiable on ]a, b[. If f(a) = f(b), then there exists a number x_{o} in ]a, b[ such that f '(x_{0}) = 0.
You have already seen in your calculus course that the derivative f '(x_{0}) at some point x_{o} gives the slope of the tangent at (x_{0}, f(x_{0}) to the curve y = f(x). Therefore the theorem states that if the end values f(a) and f(b) are equal, then there exists a point x_{0} in ]a, b[ such that the slope of the tangent at the point P(x_{0}, f(x_{0}) is zero, that is, the tangent is parallel to xaxis at that point. In fact we can have more than one point at which f(x) = 0 as shown in Figure. This shows that the number x_{0} in Theorem may not be unique.
