rolles theorem problem
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Rolle’s Theorem
Let a < b. If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) and f (a) = f (b), then there is a c in (a, b) with f 0 (c) = 0. That is, under these hypotheses, f has a horizontal tangent somewhere between a and b. Rolle’s Theorem, like the Theorem on Local Extrema, ends with f 0 (c) = 0. The proof of Rolle’s Theorem is a matter of examining cases and applying the Theorem on Local Extrema,
Some example are
1) Let f(x) = x^2. Prove that there is some c in (-2, 2) with f ' (c) = 0.2) Let . Determine if Rolle's Theorem guarantees the existence of some c in (-1, 1) with f ' (c) = 0. If not, explain why not.3) Let f(x) = x^2 – x. Does Rolle's Theorem guarantees the existence of some c in (0, 1) with f ' (c) = 0? If not, explain why not.
Let a < b. If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) and f (a) = f (b), then there is a c in (a, b) with f 0 (c) = 0. That is, under these hypotheses, f has a horizontal tangent somewhere between a and b. Rolle’s Theorem, like the Theorem on Local Extrema, ends with f 0 (c) = 0. The proof of Rolle’s Theorem is a matter of examining cases and applying the Theorem on Local Extrema,
Some example are
1) Let f(x) = x^2. Prove that there is some c in (-2, 2) with f ' (c) = 0.2) Let . Determine if Rolle's Theorem guarantees the existence of some c in (-1, 1) with f ' (c) = 0. If not, explain why not.3) Let f(x) = x^2 – x. Does Rolle's Theorem guarantees the existence of some c in (0, 1) with f ' (c) = 0? If not, explain why not.
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