The given statement is false.

Here’s the reasoning:

If $f'(c) = 0$, it means that $c$ is a critical point of the function $f(x)$. However, not all critical points correspond to a local maximum or minimum. Critical points can also correspond to:

1. Saddle points (neither a maximum nor a minimum).
2. Points of inflection (change in concavity).

For example:

• Consider $f(x) = x^3$. At $x = 0$, $f'(x) = 0$, but $x = 0$ is not a local maximum or minimum—it is an inflection point.

Would you like further clarification or detailed examples?

5 Related Questions:

1. How do you determine whether a critical point is a maximum, minimum, or saddle point?
2. What role does the second derivative test play in analyzing critical points?
3. Can a function have a critical point at $f'(c) = 0$ but still not change in value?
4. What is the geometric significance of a saddle point?
5. How can concavity determine the nature of a critical point?

Tip:

Always use the First or Second Derivative Test to classify critical points effectively.