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True or False: If the answer is false, give an example to support your answer. If (c, f(c)) is a...

Question:

True or False: If the answer is false, give an example to support your answer.

If (c, f(c)) is a critical point then f(c) is a relative extrema.

Critical Points:

Whenever a specific input value leads to the derivative of the expression containing the input variable having a value of zero, then the input value itself is considered a critical point. However, this doesn't always mean that expression has a maximum or minimum value at that input value.

Answer and Explanation:

For the given statement that "If {eq}(c,f(c)) {/eq} is a critical point then {eq}f(c) {/eq} is a relative extrema, let's analyze when the function is defined by a constant value such as {eq}f(x) = 5 {/eq}. Since critical points occur when the derivative is equal to {eq}0 {/eq}, this would mean that the function has a critical point on every input value within the function since the derivative of {eq}5 {/eq} is {eq}0 {/eq}. However, there is no relative maximum or minimum value because no matter what the input value is, the function will always have a value of {eq}5 {/eq}. For {eq}f(c) {/eq} to be a relative extrema, {eq}c {/eq} has to a critical point and the function around the input value {eq}c {/eq} has to change from a positive derivative to a negative derivative or vice versa. Also, it the function has closed bounds and the expression is not a constant, then they may be relative extrema at the endpoints of the interval. Therefore, the given statement is false.


Learn more about this topic:

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Finding Critical Points in Calculus: Function & Graph

from CAHSEE Math Exam: Tutoring Solution

Chapter 8 / Lesson 9
203K

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