Explanation Given: Let f ( x ) is differentiable. Calculation: Part a: We have to prove that f ″ ( c ) = lim h → 0 f ′ ( c + h ) h . If f ′ ( c ) = 0 is a critical point, then f ′ ( c ) = 0 and f ″ ( c ) > 0 exist. By the formula of definition of the derivative, f ″ ( c ) = lim h → 0 f ′ ( c + h ) − f ′ ( c ) h = lim h → 0 f ′ ( c + h ) − 0 h = lim h → 0 f ′ ( c + h ) h Hence, the required result

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 4.4, Problem 61E

To determine

a. Show that f″(c)=limh→0f′(c+h)h.

b. Use part (a) to show that there exists an open interval (a,b) containing c such that f′(x)<0 if a<x<c and f′(x)>0 if c<x<b. Also, conclude that f(c) is a local minimum.

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Chapter 4.4, Problem 60E

Chapter 4.4, Problem 62E

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Chapter 4 Solutions

CALCULUS (CLOTH)

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a. Show that f ″ ( c ) = lim h → 0 f ′ ( c + h ) h . b. Use part (a) to show that there exists an open interval ( a , b ) containing c such that f ′ ( x ) < 0 if a < x < c and f ′ ( x ) > 0 if c < x < b . Also, conclude that f ( c ) is a local minimum.

Solution Summary: The author explains how to determine that f′′(c)=undersethto 0mathrmlim

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a. Show that f″(c)=limh→0f′(c+h)h.

b. Use part (a) to show that there exists an open interval (a,b) containing c such that f′(x)<0 if a<x<c and f′(x)>0 if c<x<b. Also, conclude that f(c) is a local minimum.

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Chapter 4 Solutions

a. Show that f ″ ( c ) = lim h → 0 f ′ ( c + h ) h . b. Use part (a) to show that there exists an open interval ( a , b ) containing c such that f ′ ( x ) &lt; 0 if a &lt; x &lt; c and f ′ ( x ) &gt; 0 if c &lt; x &lt; b . Also, conclude that f ( c ) is a local minimum.

Solution Summary: The author explains how to determine that f′′(c)=undersethto 0mathrmlim

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a. Show that f″(c)=limh→0f′(c+h)h. b. Use part (a) to show that there exists an open interval (a,b) containing c such that f′(x)<0 if a<x<c and f′(x)>0 if c<x<b. Also, conclude that f(c) is a local minimum.

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Chapter 4 Solutions

a. Show that f ″ ( c ) = lim h → 0 f ′ ( c + h ) h . b. Use part (a) to show that there exists an open interval ( a , b ) containing c such that f ′ ( x ) < 0 if a < x < c and f ′ ( x ) > 0 if c < x < b . Also, conclude that f ( c ) is a local minimum.

a. Show that f″(c)=limh→0f′(c+h)h.

b. Use part (a) to show that there exists an open interval (a,b) containing c such that f′(x)<0 if a<x<c and f′(x)>0 if c<x<b. Also, conclude that f(c) is a local minimum.