(a) Explanation Given: f ( x ) = x 3 Calculation: The theorem 3.11 as follows: Suppose x = c is the location of a critical point of a function f with f ' ( c ) = 0 , and suppose both f and f ' ' are differentiable and f ' ' is continuous on an interval around x = c , then If f ' ' ( c ) is positive, then f has a local minimum at x = c . If f ' ' ( c ) is negative, then f has a local maximum at x = c . And If f ' ' ( c ) = 0 , then this test says that nothing about whether or not f has extremum at x = c . Inflection points of a function are the points in the domain of f at which its concavity changes. Since the sign of f ' ' measures the concavity of f , you can find the inflection points by looking for the places where f ' ' changes sign. A function increases where its derivative is positive. Taking this up one level , one can observe that the derivative function f ' is increasing where its derivative function f ' ' is positive. Therefore, you can check whether a function f is concave up or down by looking at the sign of its second derivative. Suppose the function is f ( x ) = x 3 Now f ' ( x ) = 3 x 2 − 1 f ' ( x ) = 3 x 2 And f</ (b) To determine To show : g ' ( 0 ) = 0 and g ' ' ( 0 ) = 0 while g has a local minimum at x = 0 . (c) To determine To show : h ' ( 0 ) = 0 and h ' ' ( 0 ) = 0 while h has a local maximum at x = 0 . (d) To determine To explain: how the parts (a)-(c) show that the second derivative tests does not tell anything

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Question

Chapter 3.3, Problem 10C

(a)

To determine

To find: f''(0)=0 and f''(0)=0 while f does not have alocal extremum at x=0 .

(b)

To determine

To show: g'(0)=0 and g''(0)=0 while g has a local minimum at x=0 .

(c)

To determine

To show: h'(0)=0 and h''(0)=0 while h has a local maximum at x=0 .

(d)

To determine

To explain:how the parts (a)-(c) show that the second derivative tests does not tell anything

Expert Solution & Answer

Chapter 3 Solutions

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f ' ' ( 0 ) = 0 and f ' ' ( 0 ) = 0 while f does not have alocal extremum at x = 0 .

Solution Summary: The author explains that the inflection points of a function are the points in the domain of f at which its concavity changes.

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To find: f''(0)=0 and f''(0)=0 while f does not have alocal extremum at x=0 .

To show: g'(0)=0 and g''(0)=0 while g has a local minimum at x=0 .

To show: h'(0)=0 and h''(0)=0 while h has a local maximum at x=0 .

To explain:how the parts (a)-(c) show that the second derivative tests does not tell anything

Chapter 3 Solutions