   Chapter 5.5, Problem 14E

Chapter
Section
Textbook Problem

# Find the numbers b such that the average value of f ( x ) = 2 + 6 x − 3 x 2 on the interval [ 0 , b ] is equal to 3.

To determine

To find:

The number b such that average value of f(x) on the interval 0,b is 3.

Explanation

1) Concept:

Use the mean value theorem for integrals.

2) Mean Value Theorem for Integrals

If f is continuous on a,b then there exists a number c in a,b such that

fc=fave=1b-aabfx dx

that is,

abfx dx=fcb-a

3) Given:

i) fx=2+6x-3x2

ii) fave=3

4) Calculation:

Since f is a polynomial function,

f is continuous over R.

Therefore, for any bR

f is continuouson the interval 0,b.

By using themean value theorem for integrals

there exists a number c in 0,b such that

fc=fave=1b-00bfx dx

Using the given information,

3=1b0b2+6x-3x2 dx

3b=ob2+6x-3x2 dx

By integrating,

3b=2x+6x22-3x330b

3b=

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