How to find critical points

  look for places where the derivative is zero or undefined

Step-by-step explanation:

Find the derivative. Critical points are where the derivative is zero, and where it is undefined.

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A polynomial function may or may not have points where the derivative is zero. It will never have points where the derivative is undefined (tangent lines are vertical).

A rational function, or one involving fractional exponents may have critical points where the derivative is undefined.

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For example, the function ...

  y = √(1 -x^2)

has the derivative ...

  y' = -x/√(1 -x^2)

This has a zero at x=0 (tangent is horizontal). It is undefined at x = ±1, where the denominator is zero (tangent is vertical). Those values of x tell you the critical points are (-1, 0), (0, 1), (1, 0).

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