# Inflection Points

An Inflection Point is where a curve changes from **Concave upward** to **Concave downward** (or vice versa)

So what is concave upward / downward ?

Concave upward is when the slope increases: |
||

Concave downward is when the slope decreases: |

Here are some more examples:

Learn more at Concave upward and Concave downward.

## Finding where ...

So our task is to find **where** a curve goes from concave upward to concave downward (or vice versa).

## Calculus

Derivatives help us!

The derivative of a function gives the slope.

The second derivative tells us if the slope increases or decreases.

- When the second derivative is
**positive**, the function is**concave upward**. - When the second derivative is
**negative**, the function is**concave downward**.

And the inflection point is where it goes from **concave upward** to **concave downward** (or vice versa).

### Example: y = 5x^{3} + 2x^{2} − 3x

Let's work out the second derivative:

- The derivative is
**y' = 15x**^{2}+ 4x − 3 - The second derivative is
**y'' = 30x + 4**

And **30x + 4** is negative up to x = −4/30 = −2/15, positive from there onwards. So:

**concave downward**up to x = −2/15

**concave upward**from x = −2/15 on

And the inflection point is at x = −2/15

### A Quick Refresher on Derivatives

In the previous example we took this:

y = 5x^{3} + 2x^{2} − 3x

and came up with this derivative:

y' = 15x^{2} + 4x − 3

There are **rules** you can follow to find derivatives. We used the "Power Rule":

- x
^{3}has a slope of 3x^{2}, so 5x^{3}has a slope of 5(3x^{2}) = 15x^{2} - x
^{2}has a slope of 2x, so 2x^{2}has a slope of 2(2x) = 4x - The slope of the
**line**3x is 3

Another example for you:

### Example: y = x^{3} − 6x^{2} + 12x − 5

The derivative is: y' = 3x^{2} − 12x + 12

The second derivative is: y'' = 6x − 12

And 6x − 12 is negative up to x = 2, positive from there onwards. So:

**concave downward**up to x = 2

**concave upward**from x = 2 on

And the inflection point is at x = 2: