Explanation Given: The graph has inflection point at x = c in such a way that the derivative f ' has a local minimum 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: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 3.3, Problem 8C

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 minimum at x = c

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Chapter 3.3, Problem 7C

Chapter 3.3, Problem 9C

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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 minimum 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 minimum 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 minimum 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 minimum 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 minimum at x = c