# Second derivative test

At a critical point, if f'' is positive the curve bends upward (a minimum). If f'' is negative it bends downward (a maximum).

Maths (university) · Part 04: Applications of derivatives · Card 3 of 9 · Last updated 2026-10-19

## Analogy

A bowl holds water (minimum), an upside-down bowl sheds it (maximum).

## Example

**f(x) = x^3 - 3x, f'' = 6x**

f''(-1) = -6, negative: maximum.
f''(1) = 6, positive: minimum.
f''(0) = 0 and the bend flips.
So x = 0 is an inflection point.

## In practice

Often quicker than checking signs on both sides, and f'' also tells you where the curve changes bend.

## Practice

### Flashcards

- **If f'(c) = 0 and f''(c) < 0, what is c?** A local maximum.

### Interview questions

**A rectangle has perimeter 100 m. Find its dimensions for maximum area and justify that it is a maximum.**

- Let sides be x and 50 - x
- A = 50x - x^2
- A' = 50 - 2x = 0 gives x = 25
- A'' = -2 < 0 so a maximum
- Area is 625 m^2, a square

**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 does the second derivative tell me?

It tells you how the slope is changing. A positive f'' means the slope is increasing, so the graph is concave up like a cup. A negative f'' means the slope is falling and the graph is concave down like a cap.

### What if f'' equals zero at a critical point?

The test gives no answer. The point could be a maximum, minimum or neither. Go back to the first derivative test. For x to the fourth, x cubed and negative x to the fourth, f'' is zero at 0, yet the behaviour differs in each case.

## 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)
- [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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