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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

11-14 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.

f ( x ) = x 3 3 x + 2 ,    [ 2 , 2 ]

To determine

To verify:

(i) Whether the function satisfies the hypotheses of Mean Value Theorem on the given interval.

(ii) Find all numbers c satisfy the conclusion of Mean Value theorem.

Explanation

1) Concept:

Using the Mean Value Theorem verify the result and find all the values of c

2) Theorem:

Mean value theorem- Let f be a function that satisfies the followinghypotheses:(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=x3-3x+2, [-2, 2]

4) Calculations:

(i) Consider the given function fx=x3-3x+2

As a polynomial function is always continuous and differentiable everywhere.

Therefore, f(x) is continuous on [ -2, 2] and differentiable on (-2, 2).

Hence, it satisfies hypothesis of Mean Value Theorem.

(ii) By using Mean value theorem, as f(x) is continuous on [ - 2, 2] and differentiable on

 (-2, 2)

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