First derivative test

Definition

The first derivative test is a method used to determine the local maxima and minima of a function by analyzing the sign changes of its first derivative. It helps in understanding the increasing and decreasing behavior of the function.

5 Must Know Facts For Your Next Test

1. A critical point is where the first derivative, $f'(x)$, is zero or undefined.
2. If $f'(x)$ changes from positive to negative at a critical point, then $f(x)$ has a local maximum at that point.
3. If $f'(x)$ changes from negative to positive at a critical point, then $f(x)$ has a local minimum at that point.
4. If there is no sign change in $f'(x)$ around a critical point, it implies that there is neither a maximum nor minimum at that point.
5. The first derivative test can also help determine intervals where the function is increasing or decreasing.

Review Questions

• What does it mean if $f'(x)$ changes from positive to negative at a critical point?
• How do you find the critical points of a function?
• What conclusion can be drawn if there is no sign change in $f'(x)$ around a critical point?

Related terms

Critical Point: A point on the graph of a function where its first derivative is zero or undefined.

Second Derivative Test: A method used to classify critical points as local maxima, minima, or saddle points using the second derivative.

$Increasing/Decreasing\ Function$: $A\ function\ f(x)\ is\ increasing\ on\ an\ interval\ if\ f'(x)\ >\ 0,\ and\ decreasing\ if\ f'(x)\ < \0.$