Rolle's Theorem: Rolle's theorem establishes a crucial connection between the derivative and the local behavior of a function. It states that if a function...
Rolle's Theorem:
Rolle's theorem establishes a crucial connection between the derivative and the local behavior of a function. It states that if a function is continuous on the closed interval [a, b], and differentiable in the open interval (a, b), then there exists a number c in (a, b) such that the derivative of the function is equal to the value of the function at c.
Mean Value theorem:
The mean value theorem provides a fundamental property of continuous functions on closed intervals. It states that for any function f(x) defined and continuous on the closed interval [a, b], there exists a number c in (a, b) such that the average rate of change of f(x) on the interval [a, b] is equal to the value of f'(c).
The Mean Value theorem can be interpreted geometrically as the tangent line to the graph of f(x) at the point (a, f(a)) is parallel to the chord connecting the points (a, f(a)) and (b, f(b))