EXPLORING CONCEPTS

Converse of Rolle's Theorem

Let f be continuous on [a, b] and differentiable on (a, b). If there exists c in (a, b) such that , does it follow that Explain.

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Rico Jay LaxaLv10

16 Jul 2020

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Related questions

EXPLORING CONCEPTS

Rolle's Theorem

Let f be continuous on [a, b] and differentiable on (a, b). Also, suppose that and that c is a real number in the interval (a, b) such that . Find an interval for the function g over which Rolle’'s Theorem can be applied, and find the corresponding critical number of g, where k is a constant,

(a)

(b)

(c)

Let b > 0 and f : [0, b] â R be continuous and f(b) = 0. Assume that f is differentiable on (0, b). Show: For every n â N, there exists some c â (0, b) such that cf â²(c) = ânf(c).

Hint: Apply Rolleâs theorem to the function g(x) = xⁿ f(x).

Darboux's Theorem

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