# Critical point

A value of x where f'(x) = 0 or f'(x) does not exist. Local maxima and minima can only happen at critical points.

Maths (university) · Part 04: Applications of derivatives · Card 1 of 9 · Last updated 2026-10-19

## Analogy

The top of a hill and the bottom of a valley are both flat for a moment.

## Example

**f(x) = x^3 - 3x**

f'(x) = 3x^2 - 3.
Set to 0: x^2 = 1.
Critical points at x = -1 and x = 1.
f(-1) = 2, f(1) = -2.

## In practice

Finding critical points is step one when searching for the highest or lowest value of a function.

## Practice

### Flashcards

- **How do you find critical points?** Solve f'(x) = 0 and find where f'(x) does not exist.

### 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

### Is every critical point a maximum or minimum?

No. A critical point is only a candidate. For f(x) = x cubed, the derivative 3x squared is zero at x = 0, but the function keeps rising through that point. You need a further test, such as the sign of f' around it.

### Why do maxima and minima occur at critical points?

At a smooth peak or valley the tangent line is flat, so the slope is zero. If the slope were positive or negative, you could move a little way and get a higher or lower value. Corners are the other possibility, where the derivative does not exist.

## Related concepts

- [First derivative test](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-2)
- [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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