Recall that the average value of a continuous function f on the closed interval [a, b]is given by favef (x)dx.- a.In high school, you probably learned that the average value of a collection of numbers isalways between the maximum and minimum number from the collection. The same is truefor 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) forall a in [a, b]. Use the Comparison Property for Definite Integrals to show thatfave < 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 thatm< fave < M.

We will use the definition of average value formula for the function.

(a) Suppose f(x) and g(x) are both continuous functions on [a, b] and that f(x) < g(x) for all æ 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.

(a) Suppose f(x) and g(x) are both continuous functions on [a, b] and that f(x) < g(x) for all æ 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...

See similar textbooks

Similar questions

Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.
Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f decreases on the interval -,x1 and increases on the interval x1,, then fx1 is a local maximum value.
If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
Does a Limiting Value Occur? A rocket ship is flying away from Earth at a constant velocity, and it continues on its course indefinitely. Let D(t) denote its distance from Earth after t years of travel. Do you expect that D has a limiting value?
bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.