f be a function that satisfies the following properties: f(x + y) = f(x)f(y) for all x, y e R, and f'(0) = 1. a. Explain why f is continuous at x = 0. b. Show the following two facts: that f(0) = 1, and that f(x) # 0 for all x E R. (You may prove them in either order, and use whichever you prove first to prove the second if you wish.) c. Show that f is continuous for all x E R. d. Show that f is differentiable for all x E R, and compute f'(x) in terms of x and f(x). e. In fact, f is uniquely defined by the above properties, and is a well-known function. Which is it?

f be a function that satisfies the following properties: f(x + y) = f(x)f(y) for all x, y e R, and f'(0) = 1. a. Explain why f is continuous at x = 0. b. Show the following two facts: that f(0) = 1, and that f(x) # 0 for all x E R. (You may prove them in either order, and use whichever you prove first to prove the second if you wish.) c. Show that f is continuous for all x E R. d. Show that f is differentiable for all x E R, and compute f'(x) in terms of x and f(x). e. In fact, f is uniquely defined by the above properties, and is a well-known function. Which is it?

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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