Rolle's theorem: Is one of the foundational theorems

Rolle's theorem is one of the foundational theorems in differential calculus. It is a special case of, and in fact is equivalent to, the mean value theorem, which in turn is an essential ingredient in the proof of the fundamental theorem of calculus. In calculus, Rolle's theorem states that if a differentiable function (real-valued) attains equal values at two distinct points then it must have at least one fixed point somewhere between them where the first derivative is zero. Rolle's theorem is named after Michel Rolle, a French mathematician. Rolle’s theorem is very useful in Calculus for continuous functions. In this post, we will learn Rolle’s theorem with its geometrical interpretation along with some solved examples. Rolle's theorem provides valuable insights into the behavior of differentiable functions and is widely used in calculus, analysis, and various applications in mathematics and science.