#4. Last two bullit points. Thanks. 4. A theorem states that any continuous function on a closed and bounded interval [a, b] isuniformly 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 continuouson [0, 1]. Hint: Emulate the proof of f(x) = x² is uniformly continuous on [1, 2] in thenotes. Notice that the same proof can be used to show that f(x) = x³ is uniformlycontinuous on any interval [a, b].• Show that f(x) = x³ is not uniformly continuous on [0, 0). Hint: Emulate the proof off(x) = x² 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 maximumon (a, b). Does this contradict the theorem stated above?Give an example of a function f(x) that is not bounded on [a, b). Does this contradictthe theorem stated above?• What is the max and the min of f(x) = x³ on [1,3)?

We have to give an example of a function f(x) that is not bounded on [a, b].

• 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? • Give an example of a function f(x) that is not bounded on [a, b). Does this contradict the theorem stated above? • What is the max and the min of f(x) = r³ on [1,3)?

• 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? • Give an example of a function f(x) that is not bounded on [a, b). Does this contradict the theorem stated above? • What is the max and the min of f(x) = r³ on [1,3)?

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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