# Mean value theorem

If f is continuous on [a, b] and differentiable inside, there is a point c between a and b where f'(c) equals the average slope.

Maths (university) · Part 04: Applications of derivatives · Card 5 of 9 · Last updated 2026-10-19

## Analogy

If your average speed on a trip was 60 km/h, at some moment your speedometer showed exactly 60.

## Example

**f(x) = x^2 on [1, 3]**

Average slope: (9 - 1)/(3 - 1) = 4.
Solve f'(c) = 2c = 4.
So c = 2.
The point c = 2 lies inside [1, 3].

## In practice

It underlies many proofs, such as why a function with zero derivative must be constant.

## Practice

### Flashcards

- **State the mean value theorem.** Some c in (a, b) has f'(c) = (f(b) - f(a))/(b - a).

## Questions people ask

### What does the mean value theorem say in plain words?

Somewhere between two points on a smooth curve, the tangent line is parallel to the straight line joining the two ends. In driving terms, if your average speed was 60 km/h, you must have travelled at exactly 60 km/h at some instant.

### Why does the theorem need differentiability?

If the graph has a sharp corner, the tangent may never match the average slope. The function absolute x on [-1, 2] has average slope 1/3, but its slope is only -1 or 1, never 1/3. The corner at 0 breaks the theorem.

## Related concepts

- [Critical point](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-1)
- [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)
- [Related rates](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-6)

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