Category Archives: MVT

Rudin: Ch. 5 #4

Problem Statement: If where are real constants, prove that has at least one real root between and . Proof: Consider the polynomial on . Then it follows that and which equals by our assumption. Furthermore, since is a polynomial with real coefficients … Continue reading

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April 2009 #2

Problem Statement: Suppose is a real-valued differentiable function such that for every . Show there is at most one such that . Proof: Suppose to the contrary that there are two such values such that and . Wlog assume that . Since … Continue reading

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Spring 2008 #2

Problem Statement: Let be a twice differentiable real valued function defined on . Suppose that with and . Prove there exists a such that . Proof: Since is twice differentiable on we know that is differentiable on , which implies that … Continue reading

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May 1999 #5

Problem Statement: Let be a real valued differentiable function on whose derivative is bounded on .  Show that converges. Proof: First note that for every . Furthermore we know that is bounded so there exists a real number such that for every … Continue reading

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Winter 1995 #3

Problem Statement: Let be continuous, , and exists for every . Let be increasing on . Show is increasing on . Proof: We wish to show that is increasing on which means that for . Since we know that is continuous and … Continue reading

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May 2000#2

Problem Statement: Use the Mean Value Theorem to show that for all . Proof: Let . We wish to show that for every . First note that for all . Let and consider the interval . Since is continuous and differentiable on … Continue reading

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September 2001 #1

Problem Statement: Let be differentiable on and suppose that exists and is finite. Prove if exists and is finite, then . Proof: Suppose that and that . Since these limits exist we know that these limits will exist, and be the … Continue reading

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