Suppose that f is a continuous function on [a,b]. Is the given statement true or false? If they're true, explain why. If they're false, explain why or give examples that show it's false. a) f attains an absolute maximum f(c) and an absolute minimum f(d) for some numbers c and d in [a, b] b) f'(c) Exists for every c in [a, b) c) f attains every possible value between f(a) and f(b)

College Algebra (MindTap Course List)
12th Edition
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Author:R. David Gustafson, Jeff Hughes
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Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 96E: Determine if the statement is true or false. If the statement is false, then correct it and make it...
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Deere min is statement b and c are true or false and why
**Continuity and Properties of Functions**

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**Introduction:**

Given a continuous function \( f \) on a closed interval \([a, b]\), we consider the validity of several mathematical statements related to the behavior and properties of such a function. Evaluate each statement whether it’s true or false, and provide justification or counterexamples where necessary.

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**Statements to Evaluate:**

a) \( f \) attains an absolute maximum \( f(c) \) and an absolute minimum \( f(d) \) for some numbers \( c \) and \( d \) in \([a, b]\).

b) \( f'(c) \) exists for every \( c \) in \([a, b]\).

c) \( f \) attains every possible value between \( f(a) \) and \( f(b) \).

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**Explanations:**

**a)** **Existence of Absolute Extrema on \([a, b]\)**

For any continuous function \( f \) defined on a closed and bounded interval \([a, b]\), the Extreme Value Theorem asserts that \( f \) must attain both an absolute maximum and an absolute minimum on this interval. This means there will exist points \( c \) and \( d \) in \([a, b]\) such that \( f(c) \) is the highest value and \( f(d) \) is the lowest value that \( f \) reaches on \([a, b]\).

**b)** **Existence of Derivative \( f'(c) \)**

This statement is **false**. A function \( f \) being continuous on \([a, b]\) does not guarantee that its derivative \( f'(c) \) exists everywhere on \([a, b]\). For example, the function \( f(x) = |x| \) is continuous everywhere but not differentiable at \( x = 0 \).

**c)** **Intermediate Value Property**

By the Intermediate Value Theorem, if \( f \) is continuous on \([a, b]\), then it takes every value between \( f(a) \) and \( f(b) \). This means for every value \( y \) between \( f(a) \) and \( f(b) \), there is some \( x \) in \([a, b]\) such that \( f(x) = y \).

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**Conclusion:
Transcribed Image Text:**Continuity and Properties of Functions** --- **Introduction:** Given a continuous function \( f \) on a closed interval \([a, b]\), we consider the validity of several mathematical statements related to the behavior and properties of such a function. Evaluate each statement whether it’s true or false, and provide justification or counterexamples where necessary. --- **Statements to Evaluate:** a) \( f \) attains an absolute maximum \( f(c) \) and an absolute minimum \( f(d) \) for some numbers \( c \) and \( d \) in \([a, b]\). b) \( f'(c) \) exists for every \( c \) in \([a, b]\). c) \( f \) attains every possible value between \( f(a) \) and \( f(b) \). --- **Explanations:** **a)** **Existence of Absolute Extrema on \([a, b]\)** For any continuous function \( f \) defined on a closed and bounded interval \([a, b]\), the Extreme Value Theorem asserts that \( f \) must attain both an absolute maximum and an absolute minimum on this interval. This means there will exist points \( c \) and \( d \) in \([a, b]\) such that \( f(c) \) is the highest value and \( f(d) \) is the lowest value that \( f \) reaches on \([a, b]\). **b)** **Existence of Derivative \( f'(c) \)** This statement is **false**. A function \( f \) being continuous on \([a, b]\) does not guarantee that its derivative \( f'(c) \) exists everywhere on \([a, b]\). For example, the function \( f(x) = |x| \) is continuous everywhere but not differentiable at \( x = 0 \). **c)** **Intermediate Value Property** By the Intermediate Value Theorem, if \( f \) is continuous on \([a, b]\), then it takes every value between \( f(a) \) and \( f(b) \). This means for every value \( y \) between \( f(a) \) and \( f(b) \), there is some \( x \) in \([a, b]\) such that \( f(x) = y \). --- **Conclusion:
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