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 upward slope increases
Concave downward is when the slope decreases:   concave downward slope decreases

Here are some more examples:

concave 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).

inflection points

Calculus

Derivatives help us!

The derivative of a function gives the slope.

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

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

Example: y = 5x3 + 2x2 − 3x

5x^3 2x^2 3x inflection point

Let's work out the second derivative:

 

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

f(x) is concave downward up to x = −2/15
f(x) is 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 = 5x3 + 2x2 − 3x

and came up with this derivative:

y' = 15x2 + 4x − 3

There are rules you can follow to find derivatives, and we used the "Power Rule":

Another example for you:

Example: y = x3 − 6x2 + 12x − 5

The derivative is: y' = 3x2 − 12x + 12

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

 

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

f(x) is concave downward up to x = 2
f(x) is concave upward from x = 2 on

And the inflection point is at x = 2:

x^3 6x^2 12x 5 inflection point