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Definition: A stationary point (or critical point) is a point on a curve (function) where the gradient is zero (the derivative is équal to 0). A stationary point is therefore either a local maximum, a local minimum or an inflection point.

Example: The curve of the order 2 polynomial $ x ^ 2 $ has a local minimum in $ x = 0 $ (which is also the global minimum)

Example: $ x ^ 3 $ has an inflection point in $ x = 0 $

How to calculate stationary points?

Calculate the derivative $ f' $ of the function $ f $ and look at the values for which it is canceled $ f'(x) = 0 $

If it changes sign from positive to negative, then it is a local maximum.

If it changes sign from negative to positive, then it is a local minimum.

If it does not change sign, then it is an inflection point.

A turning point is a point on the curve where the derivative changes sign so either a local minimum or a local maximum.

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