Category Archives: Differentiable

UIC Master’s Exam- Fall 2007 R2

Problem Statement:  is continuous on . Given that (a) prove has continuous derivative on (Hint: start with the change of variable u=x+t). (b) Given and show that there exists a such that for every . Solutions: (a)Proof: First let us start … Continue reading

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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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Fall 2004 #4

Problem Statement: Let be a continuous function on such that is differentiable at all points of except possibly at a single point . If exists show that exists and that . Proof: by definition. Consider: . Applying the definition stated above we … Continue reading

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Spring 2004 #5

Problem Statement: Let be differentiable. Let such that and . If is between and then there exists such that . Proof: Wlog assume . Fix . Define . Then . Note that and since . and so for sufficiently close to , … 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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