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Rolle's Theorem

The theorem is named after the seventeenth century French mathematician Michel Rolle (1652-1719).

Theorem

Let f be a continuous function defined on [a, b] and differentiable on ]a, b[. If f(a) = f(b), then there exists a number xo in ]a, b[ such that f '(x0) = 0.


You have already seen in your calculus course that the derivative f '(x0) at some point xo gives the slope of the tangent at (x0, f(x0) to the curve y = f(x). Therefore the theorem states that if the end values f(a) and f(b) are equal, then there exists a point x0 in ]a, b[ such that the slope of the tangent at the point P(x0, f(x0) is zero, that is, the tangent is parallel to x-axis at that point. In fact we can have more than one point at which f(x) = 0 as shown in Figure. This shows that the number x0 in Theorem may not be unique.
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