First Derivative Test

Core Idea

The first derivative test classifies critical points (where f'(c) = 0 or f'(c) is undefined) as local maxima, local minima, or neither. If f' changes from positive to negative at c, then f has a local maximum at c. If f' changes from negative to positive, it is a local minimum. If f' does not change sign, c is neither (like x³ at x = 0). The test works by analyzing the sign of f' on intervals determined by critical points.

How It's Best Learned

Find critical points by setting f'(x) = 0 and identifying where f' is undefined. Build a sign chart for f' across the intervals. Determine increasing/decreasing behavior. Classify each critical point by the sign change pattern.

Common Misconceptions

• Assuming f'(c) = 0 always means there is a local extremum (x³ at 0 is a counterexample).
• Forgetting to check where f' is undefined (these are also critical points).
• Only checking the sign at one point per interval instead of determining the sign throughout.

Explainer

The derivative gives the instantaneous rate of change of f at each point. When f'(x) > 0, the function is climbing left to right; when f'(x) < 0, it is falling. A critical point is any x-value where this climbing-or-falling behavior could reverse — either because f'(c) = 0 (the slope is momentarily flat) or because f'(c) is undefined (a corner or vertical tangent). The first derivative test turns this qualitative picture into a classification procedure.

Here is the core procedure. Find all critical points by computing f'(x), setting it to zero, and identifying where it is undefined. Mark these points on a number line. Pick one test point in each resulting interval and evaluate the sign of f' there. A positive sign means f is increasing in that interval; negative means decreasing. Then read off the extrema from the sign transitions: if f' goes from positive to negative at c, the function rose then fell — c is a local maximum. If f' goes from negative to positive, c is a local minimum. If the sign is the same on both sides, no extremum: c is an inflection point of the slope (the standard example is f(x) = x³ at x = 0, where f' = 3x² ≥ 0 on both sides).

The sign chart is the organizational tool that makes this systematic. Draw a number line, mark all critical points, choose a test value in each interval, compute f' at each test value, and record + or −. The chart tells you everything: increasing/decreasing behavior and the classification of each critical point, all from a single organized table. You are not computing f everywhere — you are asking a binary question (is f' positive or negative?) in each region, which is far easier.

Because you know the chain rule, you can find critical points for composite functions like f(x) = (x² − 1)^2/3. Here f' involves a factor that is undefined at x = ±1, even though f itself is defined there — those are critical points that a naive "set f' = 0" step would miss. Always hunt for *both* sources of critical points: f'(c) = 0 and f'(c) undefined. The sign chart catches all of them once you have identified them.