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Calculus : Rolle's Theorem

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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) = 0 Proof: Since f(x) is continuous on [a, b]. So it is bounded on [a, b], and attain its bounds Let M = supremum and m = infimum of [a, b] then M = m or M ? m Case 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 case a b Case 2: If M ? m, then at least one of them is diffe ...
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