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# Inflection Point

> An inflection point is a point where the graph of the function **changes** CONCAVITY (from up to down or vice versa).

It could be seen as a `Switching point`, which means the point that the `Slope` of function switch from increasing and decreasing. e.g., the function might be **still going up**, but at such a point **it suddenly increases slower and slower**. And we call that point an `inflection point`. ![image](https://user-images.githubusercontent.com/14041622/40535579-13277fba-603c-11e8-93e5-c09a446700e7.png)

[Refer to Khan academy video for more intuition rapidly: Inflection points from graphs of function & derivatives](https://www.khanacademy.org/math/ap-calculus-ab/ab-derivatives-analyze-functions/modal/v/identifying-inflection-points-from-graphs-of-function-and-derivatives)

Algebraically, we identify and express this point by the function's `First Derivative` OR `Second Derivative`. ![image](https://user-images.githubusercontent.com/14041622/40535447-b9081cec-603b-11e8-812c-89d7c77ab6c3.png)

## Example

![image](https://user-images.githubusercontent.com/14041622/40620823-de14fa26-62cc-11e8-9584-b4d2adf1b774.png) Intuitive way to solve:

* Draw a **tangent line** in imagination and move it on the function from left to right
* Notice the tangent line's slope, does it go faster or slower or suddenly change its pace at a point?
* We found it suddenly changed at point `c`.

More definitional way to solve:

* Looking for the `parts of concavity shapes`
* Seems that `B-C` is a part of `Concave Down`, and `C-D` is a part of `Concave Up`
* So `C` is a SWITCHING POINT, it's a `inflection point`.

## Example

![image](https://user-images.githubusercontent.com/14041622/40620305-e8f336b2-62ca-11e8-94b1-1d03ac18651a.png) Solve:

* Looking for the `parts of concavity shapes`
* There's no changing of concavity shapes, there's only one shape: Concave down.

## Example

![image](https://user-images.githubusercontent.com/14041622/40632882-29c5d444-631e-11e8-9bd6-986593b53471.png) Solve:

* Notice that's the graph of `f'(x)`, which is the First Derivative.
* Checking `Inflection point` from 1st Derivative is easy: just to look at the change of direction.
* Obviously there're only two points changed direction: -1 & 2

## Example

![image](https://user-images.githubusercontent.com/14041622/40632978-e07bc612-631e-11e8-96ed-7cab4ca16c84.png) Solve:

* Mind that this is the graph of `f''(x)`, which is the Second derivative.
* Checking `inflection points` from 2nd derivative is even easier: just to look at when it changes its sign, or say crosses the X-axis.
* Obviously, it crosses the X-axis 5 times. So there're 5 inflection points of `f(x)`.

## Example: Finding Inflection points

![image](https://user-images.githubusercontent.com/14041622/40635878-dd2785ae-632e-11e8-914e-73cee5899f50.png) Solve:

* Function has **POSSIBLE** inflection points when `f''(x) = 0`.
* Set `f''(x) =0` and solve for `x`, got `x=-3`. 
* We now know the possible point, but don't know its **CONCAVITY**. This need to try some numbers from its both sides:

  ![image](https://user-images.githubusercontent.com/14041622/40636008-899e2cca-632f-11e8-9d10-08f56c81631f.png)
* So it didn't change the concavity at point `-3`, means there's no inflection point for function.

## Example: Finding Inflection points

![image](https://user-images.githubusercontent.com/14041622/40636182-71f4ce5c-6330-11e8-9fc1-93e94a01c93b.png) Solve:

* Function has **POSSIBLE** inflection points when `f''(x) = 0`.
* Set `f''(x) =0` and solve for `x`, got `x=0 or 6`. 

  \[Refer to Symbolab for `f''(x)`.]\(<https://www.symbolab.com/solver/step-by-step/\frac{d^{2}}{dx^{2}}\left(3x^{5}-30x^{4}\right>))

  [Refer to Symbolab for `f''(x)=0`.](https://www.symbolab.com/solver/step-by-step/60x^{3}-360x^{2}%3D0)
* We now know the possible point, but don't know its **CONCAVITY**. This need to try some numbers from its both sides:

  ![image](https://user-images.githubusercontent.com/14041622/40636258-bfbf09ae-6330-11e8-962e-3b9f5f4d6a19.png)
* So it didn't change the concavity at point `0`, means only `6` is the inflection point.