First derivative test
The first derivative test uses the sign of a function's derivative around a critical point to classify it: if f′ changes from positive to negative, the point is a local maximum; from negative to positive, a local minimum.
The first derivative test is a method for classifying a function's critical points — the places where f′(x) = 0 or f′ is undefined — as local maxima, local minima, or neither. It works because the sign of the derivative tells you whether the function is rising or falling: f′ > 0 means increasing, f′ < 0 means decreasing.
To apply it, find the critical points, then check the sign of f′ on each side. If f′ changes from positive to negative at a critical point, the function rises then falls, so the point is a local maximum. If f′ changes from negative to positive, the function falls then rises — a local minimum. If the sign doesn't change, the point is neither; the function just pauses, as y = x³ does at x = 0.
For example, f(x) = x³ − 3x has f′(x) = 3x² − 3, giving critical points at x = ±1. A sign chart shows f′ > 0 for x < −1, f′ < 0 between −1 and 1, and f′ > 0 for x > 1. So x = −1 is a local maximum and x = 1 is a local minimum. The companion second derivative test classifies critical points using concavity instead, but the first derivative test is more general — it still works when f″ is zero or hard to compute.
The AP Calculus AB exam tests the first derivative test directly, often through sign charts or graphs of f′, and the FE Mechanical exam includes it within its differential calculus coverage of maxima and minima.
Key takeaways
- Critical points occur where f′(x) = 0 or f′ is undefined; the first derivative test classifies them.
- A sign change in f′ from positive to negative means a local maximum; negative to positive means a local minimum.
- No sign change means the critical point is neither a max nor a min (e.g., y = x³ at x = 0).
- It applies even when the second derivative test is inconclusive, making it the more general tool.