Rolle's Theorem is a fundamental concept in differential calculus that deals with continuous and differentiable functions. It states that for a function continuous on [a, b] and differentiable on (a, b) with equal endpoint values, there exists at least one point c in (a, b) where the derivative is zero. This theorem is closely related to the Mean Value Theorem and is essential for analyzing function behavior, as demonstrated through practical examples involving trigonometric and polynomial functions.
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Rolle's Theorem states that if a function is continuous, differentiable, and has equal values at the endpoints of a closed interval, then there exists at least one point within the interval where the derivative is zero
Continuity
The function must be continuous on the closed interval
Differentiability
The function must be differentiable on the open interval
Equal Values at Endpoints
The function must have equal values at the endpoints of the interval
Rolle's Theorem is a specific instance of the Mean Value Theorem, where the average rate of change is zero
The proof of Rolle's Theorem is based on the Extreme Value Theorem, which states that a continuous function on a closed interval attains both maximum and minimum values
To apply Rolle's Theorem, one must confirm the function's continuity and differentiability, as well as equal values at the endpoints, and then solve for the point where the derivative is zero
The function f(x) = cos(x) + 2 on the interval [0, 2π] is an example of Rolle's Theorem, where the derivative is zero at x = 0, π, and 2π
The function f(x) = x^3 - x on [-1, 1] is an example of Rolle's Theorem, where the derivative is zero at some point within the interval
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The theorem named after the 17th-century mathematician ______, ensures that for a continuous and differentiable function f on [a, b] with equal values at the endpoints, there is at least one point c in (a, b) where the function has a horizontal tangent, meaning f'(c) = ______.
Michel Rolle
zero
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Continuous Function Definition
A function with no interruptions or discontinuities in a given range.
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Differentiable Function Characteristic
A function with a defined derivative at every point within an open interval.
