Rolle's Theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), where f(a) = f(b), then there exists at least one number c in the open interval (a,b) where the derivative f'(c) is equal to 0. The document provides an example of applying Rolle's Theorem to the polynomial function f(x) = x^2 - 2.5x + 1.5 on the interval [1,2], showing that the derivative is 0 at x = 1.5.

1. What Goes Up,
Must Come Down
f ' (c) = 0
a b
Rolle's Theorem
http://www.youtube.com/watch?v=MSmKnUD59pA
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2. f(x) is a continuous function on the closed interval [a, b].
f(a) = f(b)
f(x) is a differentiable function on the open interval (a, b).
there exists at least one x­value, c such that f'(c) = 0
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3. Since:
1) f(x) is a polynomial function so it is continuous
on the closed interval [1, 2]
2) f(1) = f(2) = 0
3) f(x) is differentiable for all x­values on the
open interval (1, 2)
Therefore by Rolle's Theorem there exists at least one
x­value c such that f'(c) = 0. (This occurs at the vertex, x = 1.5.)
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