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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** is when the slope decreases:

Here are some more examples:

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

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} âˆ’ 3x**

Let's work out the second derivative:

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

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

f(x) is **concave downward** up to x = âˆ’2/15

f(x) is **concave upward** from x = âˆ’2/15 on

And the inflection point is at x = âˆ’2/15

In the previous example we took this:

y = 5x^{3} + 2x^{2} âˆ’ 3x

and came up with this derivative:

y' = 15x^{2} + 4x âˆ’ 3

There are **rules** you can follow to find derivatives, and 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}âˆ’ 6x^{2}+ 12x âˆ’ 5The derivative is: y' = 3x

^{2}âˆ’ 12x + 12The second derivative is: y'' = 6x âˆ’ 12

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

f(x) is

**concave downward**up to x = 2f(x) is

**concave upward**from x = 2 onAnd the inflection point is at x = 2:

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