2007-10-24 Rolle's Theorem

# 2007-10-24 Rolle's Theorem - b By hypothesis.jhas a...

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\' .l'(c) = () o x e (a) y f'(e.,) = 0 ~ o x (b) , Rolle'stheorem says that a differentiable curve has at least on.e horizontal tangent between any two points where it crosses the x-axis. It may have just one (a), or it may have more (b). ,,,. . , I .:.;c. . Rolle's Theorem Suppose that y = I(x) is continuous at every point of the closed interval [a. b] and differentiableat every point of its interior (a, b). If I(a) = I(b) = 0, then there is at least one number c in (a, b) at which F(c) =o. Proof Beingcontinuous, I assumes absolute maximum and minimum values on [a, b]. These can occur only 1. at interior points \.\'here f' is zero, 2. at interior points where f' does not exist, 3. at the endpointsof the function's domain, in this case a and

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Unformatted text preview: b. By hypothesis.jhas a derivativeat every interior point. That rules out (2), leaving us with interior points where f' = 0 and with the two endpoints a and b, If either the maximumor the minimum occurs at a point r: i:\side the interval, then f'(c) = 0 If both maximum and minimum are at a or b. then I is constant. f' = 0, and c can be taken anywhere in the interval. This completes the proof. )' tI (a) Discontinuous al an cndpoint No horizontal tangent. x y y = f(x) a (h) Disconlinuous at an interior point x y )' = f(x) It'. 1/ b (c) Continuous on [tI,//]bUI nol tlirrercnliahle al some inlcrior poinl...
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2007-10-24 Rolle's Theorem - b By hypothesis.jhas a...

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