# What are the critical numbers of f?

Finds every critical number of f, where f′ is 0 or does not exist, and which are maxima or minima.

- Page: https://www.acalculator.org/math/critical-number-calculator
- JSON spec: https://www.acalculator.org/math/critical-number-calculator.json
- Version: bc3ef8399d8c

## Default answer

Example with the default inputs (Function f(x) x^3 - 3x^2 - 9x - 1): The critical numbers of x^3 - 3x^2 - 9x - 1 are -1 (local maximum), 3 (local minimum).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Function f(x) | The function of x. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| points | Critical numbers | Where f′ is 0 or does not exist. |
| derivative | f′(x) = | The derivative of f. |

## Method

c is a critical number when f(c) is defined and f′(c) is 0 or does not exist. The sign of f′ on each side tells a maximum from a minimum.

## Assumptions

- x in radians; ln is the natural logarithm.
- Points searched for in ±10^6.

## Worked examples

1. f = x^3 - 3x^2 - 9x - 1 gives points = -1 (local maximum), 3 (local minimum). Source: OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.5 Derivatives and the Shape of a Graph, Example 4.17.
2. f = 5x^(1/3) - x^(5/3) gives points = -1 (local minimum), 0 (f′ does not exist; neither), 1 (local maximum). Source: OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.5 Derivatives and the Shape of a Graph, Example 4.18.
3. f = x^5 - 5x^3 gives points = -sqrt(3) (local maximum), 0 (neither), sqrt(3) (local minimum). Source: OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.5 Derivatives and the Shape of a Graph, Example 4.20.

## FAQ

### What is a critical number?

A number c inside the domain of f where f′(c) = 0 or f′(c) does not exist. At a critical number the graph has a horizontal tangent, a corner, or a vertical tangent. Local maxima and minima of a function can only happen at critical numbers.

### How do I find critical numbers?

Differentiate f, then solve f′(x) = 0 and find where f′ is undefined, keeping only the x where f itself is defined. For f(x) = x³ − 3x² − 9x − 1: f′(x) = 3x² − 6x − 9 = 3(x − 3)(x + 1), which is 0 at x = −1 and x = 3.

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

No. By the first derivative test, c is a local minimum if f′ changes from negative to positive at c, a local maximum if it changes from positive to negative, and neither if the sign stays the same. For x³ the critical number 0 is neither: f′ = 3x² is positive on both sides.

### Why is 0 a critical number of x^(2/3) (x − 5)?

Its derivative, (5x − 10)/(3x^(1/3)), does not exist at 0, while f(0) = 0 does. The graph has a sharp point (a cusp) there, and it is a local maximum. Points where f itself is undefined, such as 0 for 1/x, are not critical numbers.

### Why does the page give no answer for sin(x)?

Its derivative cos(x) is 0 at infinitely many points, x = π/2 + kπ. The page lists at most 30 points, and says so when there are more.

### How is the answer checked?

The derivative from the computer algebra system is compared with a numeric difference quotient at 20 points. The zeros are found both by the algebra and by a sign scan on a fine grid, so none is missed between them; a zero the algebra gives is checked by putting it back into f′.

## Sources

- OpenStax, Calculus Volume 1, section 4.3 Maxima and Minima: https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima
- OpenStax, Calculus Volume 1, section 4.5 Derivatives and the Shape of a Graph: https://openstax.org/books/calculus-volume-1/pages/4-5-derivatives-and-the-shape-of-a-graph