Suppose that a function f is continuous on [0, 1] except at 0.25 and that f (0) = 1 and f (1) = 3. Let N = 2. Sketch two possible graphs of f , one showing that f might not satisfy the conclusion of the Intermediate Value Theorem and one showing that f might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn’t satisfy the hypothesis).

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To determine To sketch: The graphs of f ( x ) which satisfy the conclusion of Intermediate Value Theorem and which does not satisfy the conclusion of Intermediate Value Theorem Explanation Given: The function f ( x ) is continuous on the interval [ 0 , 1 ] except at x = 0.25 and f ( 0 ) = 1 , f ( 1 ) = 3 and also let N = 2 . Intermediate Value Theorem: “Suppose that f is continuous on the closed interval [ a , b ] and let N be any number between f ( a ) and f ( b ) , where f ( a ) ≠ f ( b ) . Then, there exists a number c in ( a , b ) such that f ( c ) = N ”. Graph: Consider the graph of the function f ( x ) as shown below in Figure 1. From Figure 1, it is observed that f ( x ) does not satisfies the hypothesis of the Intermediate Value Theorem. That is, the function f is continuous on the closed interval [ 0 , 1 ] except at x = 0.25 . Choose N = 2 which lies between f ( 0 ) and f ( 1 ) , where f ( 0 ) ≠ f ( 1 ) . But there is no number c in ( 0 , 1 ) such that f ( c ) = 2

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Answer 1

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Single Variable Calculus

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266636

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BuyFind

Single Variable Calculus

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266636

Solutions

Chapter

Section

Answer 3

Solutions

Chapter

Section

Answer 4

Chapter

Section

Answer 5

To determine

To sketch: The graphs of f(x) which satisfy the conclusion of Intermediate Value Theorem and which does not satisfy the conclusion of Intermediate Value Theorem

Explanation of Solution

Given:

The function f(x) is continuous on the interval [0,1] except at x=0.25 and f(0)=1, f(1)=3 and also let N=2.

Intermediate Value Theorem:

“Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a)≠f(b). Then, there exists a number c in (a,b) such that f(c)=N”.

Graph:

Consider the graph of the function f(x) as shown below in Figure 1.

From Figure 1, it is observed that f(x) does not satisfies the hypothesis of the Intermediate Value Theorem. That is, the function f is continuous on the closed interval [0,1] except at x=0.25.

Choose N=2 which lies between f(0) and f(1), where f(0)≠f(1). But there is no number c in (0,1) such that f(c)=2

Single Variable Calculus

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Single Variable Calculus

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