First Derivative Test

Definition

The First Derivative Test is a method used to determine the local extrema of a function by analyzing the sign changes of its first derivative, $f'$, at critical points.

• How to read: “The first derivative of f, denoted f prime.”
• Meaning: Represents the instantaneous rate of change and the slope of the tangent line.

Why It Matters

In fields like economics or engineering, knowing whether a function is hitting a local peak or valley allows for predictable control of the system. It ensures that adjustments move the system toward an optimum rather than away from it.

Core Concepts

• Local Maximum: If $f'$ changes from positive to negative at a critical point $c$, then $f$ has a local maximum at $c$.
• Local Minimum: If $f'$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$.
• No Extremum: If $f'$ does not change sign at $c$ (e.g., positive to positive), then $f$ has no local extremum at $c$.
• The test relies on monotonicity (whether the function is strictly increasing or decreasing) to classify the extrema.

Connected Concepts

• Monotonicity
• The First Derivative Test for Local Extrema
• First Derivative Theorem for Local Extreme Values
• The Second Derivative Test for Concavity