# First derivative test

Check the sign of f' on each side of a critical point. Positive then negative means a local maximum. Negative then positive means a local minimum.

Maths (university) · Part 04: Applications of derivatives · Card 2 of 9 · Last updated 2026-10-19

## Analogy

Walking over a hill: uphill, then downhill, means you passed a peak.

## Example

**f(x) = x^3 - 3x, f' = 3(x-1)(x+1)**

x < -1: f' > 0, rising.
-1 < x < 1: f' < 0, falling.
x > 1: f' > 0, rising.
Max at x = -1, min at x = 1.

## In practice

A reliable way to classify critical points without needing the second derivative.

## Practice

### Flashcards

- **What does a sign change of f' from positive to negative mean?** A local maximum.

### Interview questions

**Find and classify the critical points of f(x) = x^3 - 3x.**

- f'(x) = 3x^2 - 3 = 0 gives x = -1 and 1
- f''(x) = 6x
- f''(-1) < 0 so local maximum, value 2
- f''(1) > 0 so local minimum, value -2

## Questions people ask

### What if f' does not change sign at a critical point?

Then it is neither a maximum nor a minimum. The function keeps going in the same direction. For f(x) = x cubed, f' is positive on both sides of 0, so x = 0 is a flat spot on the way up, not a peak or valley.

### How do I check the sign of f' between critical points?

Pick one test value in each interval and put it into f'. Only the sign matters. For 3x squared - 3, the value x = 0 gives -3, so f' is negative between -1 and 1. A continuous f' can only change sign where it is zero.

## Related concepts

- [Critical point](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-1)
- [Second derivative test](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-3)
- [Optimisation](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-4)
- [Mean value theorem](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-5)

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