First Derivative Test

First Derivative Test

The First Derivative Test is a systematic way to find local maxima and minima by analyzing how the derivative changes sign around critical numbers.

Core Principle

The derivative $f'(x)$ reveals whether a function is increasing or decreasing:

• If $f'(x) > 0$ on an interval, $f$ is increasing
• If $f'(x) < 0$ on an interval, $f$ is decreasing

Critical numbers—where $f'(c) = 0$ or $f'(c)$ is undefined—are the only places where $f'$ can change sign. These points divide the domain into intervals where $f'$ maintains constant sign.

The Test

At a critical number $c$, examine the sign of $f'$ on either side:

Step-by-Step Method

1. Find $f'(x)$
2. Solve $f'(x) = 0$ and locate where $f'(x)$ is undefined
3. Divide the domain into open intervals using these critical numbers
4. Test the sign of $f'$ at one point in each interval (build a sign chart)
5. Classify each critical number based on sign changes

Example

For $f(x) = x^3 - 3x^2 - 9x$: $f'(x) = 3x^2 - 6x - 9 = 3(x-3)(x+1)$

Critical numbers: $x = -1$ and $x = 3$

Testing signs: $f'$ is positive on $(-\infty, -1)$, negative on $(-1, 3)$, and positive on $(3, \infty)$. Therefore $f$ has a local maximum at $x = -1$ and a local minimum at $x = 3$.

Common Pitfall

Not all critical numbers are extrema. For $f(x) = x^3$, $f'(x) = 3x^2 \geq 0$ everywhere—no sign change occurs at $x = 0$, so it is neither a maximum nor minimum despite being critical. Always verify the sign actually changes.