Explanation Given Information: Theorem 3.3: A function f has local minimum at x = c . Proof: Suppose x = c is the location of local minimum of f If f ′ ( c ) does not exist, then x = c is a critical point. When f ′ ( c ) exists, then it must be equal to 0 .To show this consider the left and right hand derivatives at x = c since x = c is the location of a local minimum of f , there exist some ∂ > 0 such that for all x ∈ ( c − ∂ , c + ∂ ) , f ( x ) ≥ f ( c ) , and thus f ( x ) − f ( c ) ≥ 0 .In the case where x ∈ ( c , c + ∂ ) , it follows that x > c , which means that x − c is positive

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Chapter 3.1, Problem 64P

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To Prove: If a function fhas local minimum at x=c,then either f′(c)does not existor f′(c)=0

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Chapter 3.1, Problem 63A

Chapter 3.1, Problem 65P

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To Prove: If a function f has local minimum at x = c , then either f ′ ( c ) does not existor f ′ ( c ) = 0

Solution Summary: The author explains that if x=c is the location of a local minimum, then it must be equal to 0.

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To Prove: If a function fhas local minimum at x=c,then either f′(c)does not existor f′(c)=0

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Chapter 3 Solutions