### Blogroll

### Archives

- August 2011 (16)
- June 2011 (5)
- May 2011 (30)
- April 2011 (4)

### Categories

Absolute Convergence Algebra Analysis Cauchy Center Compact Comparison Test Continuity Countable Cyclic Dense Differentiable Field Group Integral Domain Linear Operator Math MVT Order Principal Ideal Riemann Integrable Sequence Sequence of Functions Series Series of Functions Squeeze Theorem Topology Uncategorized Uniform Continuity Uniform Convergence

# 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

Posted in Analysis, Differentiable, Math, MVT
Leave a comment

## 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

Posted in Analysis, Continuity, Differentiable, Math, MVT
Leave a comment

## 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

Posted in Analysis, Continuity, Differentiable, Math, MVT
Leave a comment

## 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

Posted in Analysis, Cauchy, Differentiable, Math, MVT
Leave a comment

## 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

Posted in Analysis, Continuity, Differentiable, Math, MVT
Leave a comment

## 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

Posted in Analysis, MVT
Leave a comment