# Where does the concavity of f change?

Finds where f is concave up or down, and its inflection points.

- Page: https://www.acalculator.org/math/inflection-point-calculator
- JSON spec: https://www.acalculator.org/math/inflection-point-calculator.json
- Version: a6062c1636b6

## Default answer

Example with the default inputs (Function f(x) x^3 - 6x^2 + 9x + 30): The inflection points of x^3 - 6x^2 + 9x + 30 are 2.

## Inputs

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

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| points | Inflection points at x = | Where f is continuous and f″ changes sign. |
| up | Concave up on | Where f″ > 0. |
| down | Concave down on | Where f″ < 0. |
| second | f″(x) = | The second derivative. |

## Method

Concave up where f″ > 0, down where f″ < 0; an inflection point is where f is continuous and f″ changes sign.

## Assumptions

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

## Worked examples

1. f = x^3 - 6x^2 + 9x + 30 gives points = 2, up = (2, ∞), down = (-∞, 2). Source: OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.5 Derivatives and the Shape of a Graph, Example 4.19.
2. f = x^4 gives points = none, up = (-∞, ∞), down = none.

## FAQ

### What is concavity?

f is concave up on an interval when its graph bends upward, like a cup (f′ is increasing there), and concave down when it bends downward, like a cap (f′ is decreasing). With a second derivative: concave up where f″ > 0, concave down where f″ < 0.

### What is an inflection point?

A point of the graph where f is continuous and the concavity changes, from up to down or from down to up. For x³ − 6x² + 9x + 30, f″ = 6x − 12 changes sign at x = 2, so (2, 32) is an inflection point.

### Is every zero of f″ an inflection point?

No. f″ must change sign. For x⁴, f″ = 12x² is 0 at x = 0 but positive on both sides, so x⁴ is concave up everywhere and has no inflection point. And an inflection point can be where f″ does not exist, as for the cube root of x at 0.

### How do I find the intervals of concavity?

Find f″, then the x where f″ = 0 or f″ does not exist, and the points where f is not defined. They cut the number line into intervals; in each, the sign of f″ at any one point gives the concavity of the whole interval.

### Why does 1/x have no inflection point?

1/x is concave down on (−∞, 0) and concave up on (0, ∞), but 0 is not in its domain: the graph does not pass through a point there, so there is no inflection point.

### How is the answer checked?

Both derivatives from the computer algebra system are compared with numeric difference quotients at 20 points. The zeros and breaks of f″ are found both by the algebra and by a scan on a fine grid, and the sign of f″ is read on each side of each point.

## Sources

- 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