   Chapter 3.2, Problem 11E

Chapter
Section
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 ) = 2 x 2 − 3 x + 1 ,    [ 0 , 2 ]

To determine

To verify:

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

(ii) Find all numbers c that 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 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=2x2-3x+1, [0, 2]

4) Calculations:

(i) Consider the given function fx=2x2-3x+1

As a polynomial function is always continuous and differentiable everywhere

Therefore, f(x) is continuous on [0, 2] and differentiable on (0, 2).

Hence, it satisfies hypothesis of Mean Value Theorem.

(ii) By using Mean Value theorem, as f(x) is continuous on [0, 2] and differentiable on (0, 2)

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