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Chapter 4: Problem 10

Suppose \(f\) is continuous on an interval containing a critical point \(c\) and \(f^{\prime \prime}(c)=0 .\) How do you determine whether \(f\) has a local extreme value at \(x=c ?\)

Short Answer

Expert verified
Explain your reasoning. Answer: We cannot determine whether f has a local extreme value at x=c by only considering the second derivative since f''(c) = 0 is inconclusive. To determine if f has a local extreme value at x=c, we need to analyze the behavior of f'(x) in a neighborhood of x=c. If f'(x) changes its sign (from positive to negative or vice versa) as x crosses c, then f has a local extreme value at x=c. If not, there is no local extreme value at x=c.

Step by step solution

01

Identify the critical point

In this problem, we are given that f has a critical point at x=c. Therefore, we will focus our analysis on this point.
02

Apply the Second Derivative Test

Since f''(c) = 0, which is inconclusive by the second derivative test, we cannot determine whether f has an extreme value at x=c by only using information about the second derivative. Instead, we need to analyze the behavior of the first derivative f'(x) in a neighborhood of x=c.
03

Analyzing the first derivative in a neighborhood of x=c

To determine if f has a local minimum, maximum or no extreme value at x=c, we need to look at how the first derivative behaves in the interval around x=c. Consider an interval (a, b) containing x=c. 1. If f'(x) changes its sign (positive to negative or vice versa) at x=c, then f has a local extreme value at x=c. 2. If f'(x) does not change its sign when passing through x=c, then f does not have a local extreme value at x=c. To check the behavior of f'(x), we can use the sign chart of f'(x) by choosing points close enough to x=c on both sides.
04

Conclusion

By analyzing the first derivative's behavior in a neighborhood of x=c, we can draw conclusions about the presence or absence of a local extreme value for the function f at x=c. If f'(x) changes its sign (from positive to negative or vice versa) as x passes through x=c, then f has a local extreme value at x=c. If not, there is no local extreme value at x=c.

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