> For the complete documentation index, see [llms.txt](https://solomons-mathbook.gitbook.io/calculus-basics/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://solomons-mathbook.gitbook.io/calculus-basics/comment-389389887/comment-391300278.md).

# Critical points

[Refer to PennCalc Main/Optimization](http://calculus.seas.upenn.edu/?n=Main.Optimization)

For analyzing a function, it's very efficient to have a look at its `Critical points`, which could be classified as `Extrema`, `Inflection`, `Corner`, and `Discontinuity`.

![image](https://user-images.githubusercontent.com/14041622/40529990-a0dc454c-6029-11e8-9ea0-d1c77536f227.png)

## `How to find critical points`

Strategy:

* Knowing that `f(x)` has critical point `c` when `f'(c) = 0` or `f'(c) is undefined`
* Differentiate `f(x)` to get `f'(x)`
* Solve `c` for `f'(c)=0 & f'(c) undefined`

\[Refer to Symbolab's step-by-step solution.]\([https://www.symbolab.com/solver/step-by-step/critical points%2C f\left(x\right)%3Dx\cdot sqrt\left(4-x\right](https://www.symbolab.com/solver/step-by-step/critical%20points%2C%20f/left\(x/right\)%3Dx/cdot%20sqrt/left\(4-x/right)))

### Example

![image](https://user-images.githubusercontent.com/14041622/40605709-d4ec9d1e-6295-11e8-9706-4348337b8bb6.png) Solve:

* See that original function `f(x)` is undefined at `x = 2 or -2`
* Differentiate `f(x)` to get `f'(x)`:

  ![image](https://user-images.githubusercontent.com/14041622/40605772-02625298-6296-11e8-8626-02696b7ddbcb.png)
* Solve `f'(x)=0` only when `x=0`.
* `f'(x)` is undefined when `x=2 or -2`, as the same with `f(x)` so it's not a solution.

### Example

![image](https://user-images.githubusercontent.com/14041622/40605182-10435260-6294-11e8-8dbf-96b12ef0be19.png) Solve: \[Refer to Symbolab step-by-step solution.]\([https://www.symbolab.com/solver/step-by-step/critical points%2C f\left(x\right)%3Dx\cdot sqrt\left(4-x\right](https://www.symbolab.com/solver/step-by-step/critical%20points%2C%20f/left\(x/right\)%3Dx/cdot%20sqrt/left\(4-x/right)))

* Differentiate `f(x)` to get `f'(x)`:

  ![image](https://user-images.githubusercontent.com/14041622/40605364-c42d5c30-6294-11e8-84e8-c4026d2550f2.png)
* `f'(x)` is **undefined** when `x > 4`
* Solve `f'(x)=0` get `x = 8/3`
* So under the given condition, only `x=8/3` is the answer.


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