Rolle's Theorem

Definition

Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on an open interval (a, b), and has equal values for its endpoints, then there exists at least one point c in (a, b) where the derivative equals zero.

Related terms

Continuity:

Continuity refers to whether or not there are any breaks or jumps in a function. For Rolle's theorem to apply, it requires continuity on the closed interval [a, b].

Differentiability:

A function is differentiable if it has a derivative at every point in its domain. Rolle's theorem requires the function to be differentiable on the open interval (a, b).

Extreme Value Theorem:

The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], then it must have both a maximum and minimum value within that interval.