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Unformatted text preview: f : S 7 R such that f does not have a maximum value on S . 5. Suppose that f : R 7 R and g : R 7 R are dierentiable functions. Prove that the dierence f-g is dierentiable. Bonus. Prove that the product f g is dierentiable. 6. a. Suppose that f : ( a, b ) 7 R is a decreasing function. Prove that for every c ( a, b ) either f is not dierentiable at c , or f ( c ) 0. b. Suppose that f : ( a, b ) 7 R is a dierentiable function and and for all c ( a, b ), f ( c ) < 0. Prove that f is decreasing. Bonus. Give an example of a continuous function f : R 7 R that is dierentiable everywhere, and such that f (0) < 0, but f is not decreasing on any open interval containing 0. Prove that your example has indeed all the required properties....
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This note was uploaded on 04/02/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.
- Spring '07