Calculus : Rolle's Theorem

Feb 3rd, 2012
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State Rolle's theorem. Prove that if "f" is continuous on [a,b] and derivable on (a,b) and f(a)=f(b), then there exists at least one point c ϵ ( a, b ) such that " f ′(c) = 0 "

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"Rolle's Theorom"Statement: Let a function f be(1) Continuous on the closed interval [ a, b ](2) Derivable in the open interval [ a, b ](3) f(a) = f(b)Then there exists at least one point c ? ( a, b ) such that f'(c) = 0Proof: Since f(x) is continuous on [a, b]. So it is bounded on [a, b], and attain its boundsLet M = supremum and m = infimum of [a, b] then M = m or M ? mCase 1: If M = m then f(x) is constant in the interval [a, b],as shown in figure f'(c) = 0 ? x ? [ a, b ] f(b)Hence, theorem is proved in this caseabCase 2: If M ? m, then at least one of them is diffe

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