Let a, b be real numbers for which b> a, and let f be a function with domain of (a, b) that is continuous at every point on its domain. (a) If e is a point on (a, b) for which f'(e) is positive, then f(e) can be neither a maximum nor a minimum value of function f. Through a combination of pictures and/or words, provide an intuitive explanation to support this statement. (b) The result in part (a) would be similar if f'(c) were negative. Explain. (c) It follows that f will only attain its maximum or minimum values at one of the three following types of points: (1) At endpoints z = a and z = b; (2) At points e on (a, b) where r(e) equals zero; At points e on (a, b) where r(e) does not exist. Explain.

Let a, b be real numbers for which b> a, and let f be a function with domain of (a, b) that is continuous at every point on its domain. (a) If e is a point on (a, b) for which f'(e) is positive, then f(e) can be neither a maximum nor a minimum value of function f. Through a combination of pictures and/or words, provide an intuitive explanation to support this statement. (b) The result in part (a) would be similar if f'(c) were negative. Explain. (c) It follows that f will only attain its maximum or minimum values at one of the three following types of points: (1) At endpoints z = a and z = b; (2) At points e on (a, b) where r(e) equals zero; At points e on (a, b) where r(e) does not exist. Explain.

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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