   Chapter 3.2, Problem 6E

Chapter
Section
Textbook Problem

# 5-8 Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. f ( x ) = x 3 − 2 x 2 − 4 x + 2 ,    [ − 2 , 2 ]

To determine

To verify:

(i) Whether the function satisfies the hypothesis of Rolle’s Theorem on the given interval.

(ii) Find all numbers c that satisfy the conclusion of Rolle’s theorem.

Explanation

1) Concept:

Using the Rolle’s Theorem verify the result and find c

2) Theorem:

Rolle’s Theorem – Let f  be a function that satisfies the following 3 hypotheses:

i. f is continuous on the closed interval [a, b]

ii. f is differentiable on the open interval (a, b)

iii. f (a)=f (b)

Then there is a number c in (a, b) such that f'c=0

3) Given:

fx=x3-2x2-4x+2, [-2, 2]

4) Calculation:

(i) Consider the given function, fx=x3-2x2-4x+5

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 i) and ii) of Rolle’s Theorem.

Now, fx=x3-2x2-4x+2

At x=-2, f-2=(-2)3-2-22-4-2+2=-8-8+8+2=-6

At x=2, f2=(2)3-222-42+2

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