First Derivative Test

Sign Chart for Increasing and Decreasing

Download PNG Save to Pinterest

Related Tools

Related Labs

Related Worksheets

Related Cheat Sheets

The First Derivative Test is a method for deciding whether a critical point is a local maximum, local minimum, or neither. It uses the sign of the derivative $f'(x)$ on intervals around a critical point instead of relying only on the graph shape. This matters because it connects slope behavior to function behavior in a precise and testable way. It is one of the main tools students use to analyze graphs in calculus.

A critical point occurs where $f'(c) = 0$ or where $f'(c)$ does not exist, as long as $c$ is in the domain of $f$ . To apply the test, find intervals on each side of the critical point and determine whether $f'(x)$ is positive or negative there. If $f'(x)$ changes from positive to negative, the function changes from increasing to decreasing, so there is a local maximum. If $f'(x)$ changes from negative to positive, the function changes from decreasing to increasing, so there is a local minimum.

Key Facts

Critical points occur where $f'(c) = 0$ or $f'(c)$ does not exist, with $c$ in the domain of $f$ .
If $f'(x) > 0$ on an interval, then $f(x)$ is increasing on that interval.
If $f'(x) < 0$ on an interval, then $f(x)$ is decreasing on that interval.
If $f'(x)$ changes from $+$ to $-$ at $x = c$ , then $f(c)$ is a local maximum.
If $f'(x)$ changes from $-$ to $+$ at $x = c$ , then $f(c)$ is a local minimum.
If $f'(x)$ does not change sign at $x = c$ , then $x = c$ is not a local extremum by the First Derivative Test.

Vocabulary

Derivative: The derivative $f'(x)$ measures the instantaneous rate of change or slope of the function at $x$ .
Critical point: A critical point is a value $c$ in the domain where $f'(c) = 0$ or $f'(c)$ does not exist.
Local maximum: A local maximum is a point where the function value is greater than nearby function values.
Local minimum: A local minimum is a point where the function value is less than nearby function values.
Sign chart: A sign chart is a diagram that shows whether $f'(x)$ is positive or negative on intervals.

Common Mistakes to Avoid

Assuming every point where $f'(c) = 0$ is a max or min, because some critical points have no sign change and are neither. Always test the sign of $f'(x)$ on both sides.
Forgetting that points where $f'(c)$ does not exist can still be critical points, because the derivative being undefined does not remove the point from consideration.
Using the function value instead of the derivative sign, because the First Derivative Test depends on whether slopes are positive or negative. Build a sign chart for $f'(x)$ , not for $f(x)$ .
Testing only one side of a critical point, because a local extremum depends on how the derivative behaves before and after the point. You need intervals on both sides to classify it correctly.

Practice Questions

1 Let $f'(x) = (x - 1)(x - 5)$ . Find the critical points and use the First Derivative Test to classify each one.
2 A function has derivative $f'(x) = x(x + 2)^2$ . Find all critical points and determine where the function is increasing, decreasing, and whether each critical point is a local extremum.
3 A function has a critical point at $x = 3$ , and $f'(x)$ is negative on both sides of $x = 3$ . What does the First Derivative Test say about $x = 3$ , and why?