AP Calculus AB and BC glossary
Rolle's Theorem
Rolle's Theorem says that if a function is continuous on a closed interval, differentiable inside it, and takes the same value at both endpoints, then there is an interior point where the derivative is zero. It is the Mean Value Theorem with equal endpoint values.
When the average rate of change is zero, so the Mean Value Theorem promises a point where . That is exactly a horizontal tangent somewhere between the two endpoints.
The standard use is proving that a function has a critical point between two roots, which in turn bounds how many roots the function can have.
Appears in: Unit 5: Analytical Applications