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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

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) = 2x2 − 4x + 5, [−1, 3]

To determine

To verify: Whether the function f(x)=2x24x+5 satisfies the Rolle’s theorem on the interval [1,3] and to find all numbers c of the function f(x)=2x24x+5 on the interval [1,3] that satisfies the conclusion of Rolle’s Theorem.

Explanation

Given:

The function f(x)=2x24x+5 is a polynomial.

Rolle’s Theorem:

“Let f be a function that satisfies the following hypothesis:

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

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

3. f(a)=f(b) .

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

Verification:

1. Since the function f(x)=2x24x+5 is polynomial function, it is continuous on .

Therefore, the function f(x) is continuous on the closed interval [1,3] .

2. Since the function f(x)=2x24x+5 is a polynomial function, it isdifferentiable everywhere.

Therefore, the function f(x) is differentiable on the open interval (1,3) .

3. To satisfy the third condition of the theorem, f(a) must be equal to f(b) .

From the given interval, it is observed that a=1 and b=3

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