Chapter 4, Problem 20RE

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Using the Mean Value Theorem(a) For the function f ( x ) = A x 2 B x + C , determine the value of c guaranteed by the Mean Value Theorem on the interval [ x 1 , x 2 ] .(b) Demonstrate the result of part (a) for f ( x ) = 2 x 2 3 x + 1 on the interval [0,4].

(a)

To determine

To calculate: The value of c such that f'(c)=f(b)f(a)ba with the help of Mean Value Theorem for f(x)=Ax2+Bx+C.

Explanation

Given:

The polynomial f(x)=Ax2+Bx+C in the interval [x1,x2].

Formula used:

The Mean Value Theorem states that if a function g(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) then for any câˆˆ(a,b),

g'(c)=g(b)âˆ’g(a)bâˆ’a

Calculation:

Consider the provided function:

f(x)=Ax2+Bx+C

The provided function is continuous and differentiable on the whole number line.

Thus Mean Value Theorem is applicable.

Compute the expression from the Mean Value theorem using the end-points provided.

f'(c)=f(x2)âˆ’f(x1)x2âˆ’x1=

(b)

To determine

To calculate: The value of c such that f'(c)=f(b)f(a)ba with the help of Mean Value Theorem for f(x)=2x23x+1.

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