Critical number

Definition

A critical number of a function is a number $c$ in the domain of the function where either the derivative is zero or does not exist. Critical numbers are used to find local maxima and minima.

5 Must Know Facts For Your Next Test

1. A critical number is found by solving $f'(x) = 0$ or determining where $f'(x)$ does not exist.
2. Critical numbers must be within the domain of the original function.
3. To classify a critical number as a maximum, minimum, or neither, you can use the First Derivative Test or the Second Derivative Test.
4. Not all critical numbers correspond to local extrema; some may be points of inflection.
5. The endpoints of a closed interval should also be checked when finding absolute maxima and minima.

Review Questions

• What is the definition of a critical number?
• How do you determine if a given value is a critical number for a function?
• What tests can be used to classify critical numbers as local maxima or minima?

Related terms

Local Maximum: A point where the function value is higher than all nearby points.

Local Minimum: A point where the function value is lower than all nearby points.

First Derivative Test: A method for determining whether a critical point is a local maximum, minimum, or neither by analyzing changes in sign of $f'(x)$.