   Chapter 5.2, Problem 54E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must ∫ 0 2 f ( x ) d x lie? Which property of integrals allows you to make your conclusion?

To determine

To find: The interval for the integral function 02f(x)dx lies between absolute minimum (m) and maximum value (M).

To identify: The property of the integral used to obtain the solution.

Explanation

Given information:

The integral function is 02f(x)dx.

Show comparison property 8 of integrals:

If mf(x)M for axb, then m(ba)abf(x)dxM(ba) (1)

Calculate:

Consider the function f(x).

The function f(x) has lower limit a=0 and upper limit b=2.

The integral function has absolute maximum value as M and absolute minimum value as m.

Modify the function f(x) as shown below:

mf(x)Mfor0x2 (2)

Compare Equation (2) with Equation (1).

The value of the function lies between its absolute maximum and absolute minimum value within limits [0,2]

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