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A critical point is an interior point of the domain of a function f where

f primef′

is zero or undefined.

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You're right! A critical point of a function \( f(x) \) occurs at an interior point in its domain where the first derivative, \( f'(x) \), is either zero or undefined. These points are important because they help identify potential local maxima, minima, or inflection points in the graph of the function.

In other words, for a critical point \( x = c \):

- \( f'(c) = 0 \) (the slope of the tangent is zero) or
- \( f'(c) \) does not exist (the derivative is undefined).

However, it’s also essential to check whether these points are within the domain of the function because critical points only apply to points inside the domain.

Would you like more details or have any questions?

Here are 5 related questions that expand on this topic:
1. How do you determine whether a critical point is a local maximum, minimum, or neither?
2. What is the difference between critical points and inflection points?
3. Can a critical point occur at the boundary of a function's domain?
4. How do you find critical points for multivariable functions?
5. Why might a derivative be undefined at a critical point?

**Tip**: Always check the second derivative \( f''(x) \) after finding critical points to confirm the nature of the point (concave up or down).

A critical point is an interior point of the domain of a function f where f prime (f') is zero or undefined.

Explore critical points in calculus, where the first derivative f'(x) equals zero or is undefined. This concept helps identify local maxima, minima, and inflection points. Learn key formulas and theorems such as the first derivative test. Suitable for Grades 11-12.

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