Critical points

Refer to PennCalc 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.

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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))

Example

Solve:

• See that original function f(x) is undefined at x = 2 or -2

• Differentiate f(x) to get f'(x):

  

• 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

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))

• Differentiate f(x) to get f'(x):

  

• 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.