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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(The Second Derivative Test) Let f : [a, b] → R be differentiable on (a, b). Suppose c ∈ (a, b) is such that f '(c) = 0, and f ''(c) exists. (a) If f ''(c) > 0, prove that f has a local minimum at c. (b) If f ''(c) < 0, prove that f has a local maximum at c. (c) Show, using two specific examples, that no conclusion can be made if f ''(c) = 0.

prove the following version of the First Derivative Test: If f′ is continuous on the interval [a,b] and if f has exactly one critical point c then f has a maximum at c if f′(a′)>0 and f′(b′)<0 for some a′ and b′ such that a<a′<c<b′<b

let f be a function with f(0)=1 and f(x)=cos(x2) a) find the linearization of f at x=0 b) estimate the falue of f at x=0.1 c) do you think the actual value of f at x=0.1 is greater or less than the estimate in part b? explain

Chapter 4 Solutions

CALCULUS (CLOTH)

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

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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.