Recall that the average value of a continuous function f on the closed interval [a, b] is given by 1 fave f(x)d.x. b - a In high school, you probably learned that the average value of a collection of numbers is always between the maximum and minimum number from the collection. The same is true for functions! The goal of this problem is to justify this. (a) Suppose f (x) and g(x) are both continuous functions on [a, b] and that f(x) < g(x) for all x in [a, b). Use the Comparison Property for Definite Integrals to show that fave < Jave. (b) Suppose that minimum value of f(x) on [a, b] is m, and the maximum value of f(x) on [a, b] is M, i.e. fmin = m and fmax = M. Use part (a) to show that m < fave < M.

Recall that the average value of a continuous function f on the closed interval [a, b] is given by 1 fave f(x)d.x. b - a In high school, you probably learned that the average value of a collection of numbers is always between the maximum and minimum number from the collection. The same is true for functions! The goal of this problem is to justify this. (a) Suppose f (x) and g(x) are both continuous functions on [a, b] and that f(x) < g(x) for all x in [a, b). Use the Comparison Property for Definite Integrals to show that fave < Jave. (b) Suppose that minimum value of f(x) on [a, b] is m, and the maximum value of f(x) on [a, b] is M, i.e. fmin = m and fmax = M. Use part (a) to show that m < fave < M.

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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