4. A theorem states that any continuous function on a closed and bounded interval [a, b] is uniformly continuous on [a, b]. Another theorem states that a continuous function on [a, b] must be bounded, and must have a max and a min on [a, b]. • Use the definition of uniform continuity to show that f(x) = x³ is uniformly continuous on (0, 1]. Hint: Emulate the proof of f(x) = x² is uniformly continuous on [1,2] in the notes. Notice that the same proof can be used to show that f(x) = x³ is uniformly continuous on any interval [a, b]. • Show that f(x) = x³ is not uniformly continuous on [0, 0). Hint: Emulate the proof of f(x) = a² is not uniformly continuous on [0, 0) in the notes. • Give an example of a continuous function f(x) on (a, b) such that f(x) has no maximum on (a, b). Does this contradict the theorem stated above?

4. A theorem states that any continuous function on a closed and bounded interval [a, b] is uniformly continuous on [a, b]. Another theorem states that a continuous function on [a, b] must be bounded, and must have a max and a min on [a, b]. • Use the definition of uniform continuity to show that f(x) = x³ is uniformly continuous on (0, 1]. Hint: Emulate the proof of f(x) = x² is uniformly continuous on [1,2] in the notes. Notice that the same proof can be used to show that f(x) = x³ is uniformly continuous on any interval [a, b]. • Show that f(x) = x³ is not uniformly continuous on [0, 0). Hint: Emulate the proof of f(x) = a² is not uniformly continuous on [0, 0) in the notes. • Give an example of a continuous function f(x) on (a, b) such that f(x) has no maximum on (a, b). Does this contradict the theorem stated above?

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 98E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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