Explanation Given: The graph has inflection point at x = c in such a way that the derivative f ' has a local maximum at x = c Calculation: Inflection points are those points in the domain of the function ‘f’ at which the concavity of the function changes. Since, the sign of double derivative f ' ' measures the concavity of the function, the inflection points can be found by looking for the places where the double derivative f ' ' changes its sign. Suppose x = c is the location of the critical point of a function f with and suppose both f and f ' are differentiable and f ' ' is continuous on an interval around x = c, then If f ' ' is positive, then function f has a local minimum at x = c If f ' ' is negative, then function f has a local maximum at x = c If f ' ' = 0 , then this test says nothing about whether or not the function f has extremum at x = c A function f is always increasing when the derivative f ' of that function is positive. In the similar manner, a derivative f ' is increasing if the double derivative f ' ' is positive. Therefore, it can be checked whether the function 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 and f ' ' ( x ) = − 6 x Here, f ' ' ( 0 ) = 0 only when x = 0

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 7C

To determine

The graph of the given function

The tangent lines to the graph of function ‘f’ to prove that the derivative f' does have a local maximum at x = c

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Chapter 3.3, Problem 6C

Chapter 3.3, Problem 8C

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The graph of the given function The tangent lines to the graph of function ‘f’ to prove that the derivative f ' does have a local maximum at x = c

Solution Summary: The author analyzes the inflection points in the domain of the function ‘f’ at which the concavity changes.

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The graph of the given function

The tangent lines to the graph of function ‘f’ to prove that the derivative f' does have a local maximum at x = c

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Chapter 3 Solutions

The graph of the given function The tangent lines to the graph of function ‘f’ to prove that the derivative f ' does have a local maximum at x = c

Solution Summary: The author analyzes the inflection points in the domain of the function ‘f’ at which the concavity changes.

Concept explainers

Videos

The graph of the given function The tangent lines to the graph of function ‘f’ to prove that the derivative f' does have a local maximum at x = c

Want to see the full answer?

Chapter 3 Solutions

The graph of the given function

The tangent lines to the graph of function ‘f’ to prove that the derivative f' does have a local maximum at x = c