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State whether the following statement is true or false. If it is false, correct the statement and...

Question:

State whether the following statement is true or false. If it is false, correct the statement and explain why it is not true, or give a counter example.

If {eq}f'(c) = 0 {/eq}, then {eq}f(x) {/eq} has a local maximum or minimum at {eq}c {/eq}

Critical Points

When the derivative is equal to zero, a function has a critical point. While these are the points where the function could have a local maximum or minimum, neither is guaranteed.

Answer and Explanation:

This is a false statement, and we can construct a fairly simple counterexample to prove it. Let's consider the function {eq}f(x) = x^3 {/eq} and show that its critical point is neither a local maximum nor a local minimum. First, let's differentiate this function using the power rule.

{eq}f'(x) = 3x^2 {/eq}


Next, we can find the critical point of this function by setting the derivative equal to zero.

{eq}3x^2 = 0\\ x = 0 {/eq}


If this is a local maximum or minimum, the first derivative will change sign on either side of this critical point. However, we will find that this function does not, in fact, change sign on either side of this point.

{eq}f'(-1) = 3(-1)^2 = 3\\ f'(1) = 3(1)^2 = 3 {/eq}


The derivative is positive on both sides, meaning that the function is increasing on both sides. Therefore, the function does not have a local minimum nor a local maximum. In fact, further analysis of this point will show that it is instead an inflection point.


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
205K

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