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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?

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### 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.

Determine whether the statement is true or false: If f'(c) = 0, then f has a local maximum or minimum at c.

Explore whether a function has a local maximum or minimum at a critical point where f'(c) = 0. Understand the role of the first and second derivative tests in identifying local extrema or inflection points. Suitable for high school calculus students.

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