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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Let f ( x ) = ( x 3 ) 2 . Show that there is no value of c in (1, 4) such that f ( 4 ) f ( 1 ) = f ( c ) ( 4 1 ) . Why does this not contradict the Mean Value Theorem?

To determine

To show:

There is no value of c in (1, 4) Such that f4-f1=f'c(4-1) and why does this not contradict theMean Value Theorem.

Explanation

1) Concept:

Using the Mean Value Theorem verify the results

2) Theorem:

Mean Value Theorem: Let f be a function that satisfies the following hypotheses:(i) f  is continuous on the closed interval [a, b].(ii) f is differentiable on the open interval (a, b). Then there is a number c in (a, b) such that f'c=fb -f(a)b - a or equivalently, f(b)  f(a) = f(c) (b  a).

3) Given:

fx=x-3 -2 on (1, 4)

4) Calculations:

(i)

As fx=x-3 -2

f'c=f4 -f(1) 4 -1

f4=4-3-2=1

f1=1-3-2=-2-2=1-2

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