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How do I distinguish between a point of local maxima or minima, when the second derivative in that point is equal to zero?

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- Thread starter greg_rack
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In summary, when determining whether a point is a local maxima or minima, a "stationary point" means that the first derivative is zero and the second derivative is also zero. If the first nonzero derivative is odd, the point is a stationary inflection point and if it is even, the point is a minimum if the derivative is positive and a maximum if it is negative. This may seem counter-intuitive, but can be demonstrated analytically by looking at the behavior of the function in a small enough neighborhood around the point. In this neighborhood, the function behaves like a power function, with the higher order terms being dominated by the nth term.

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In that case, if the first nonzero derivative is odd (third, fifth etc.), it is neither maximum nor minimum but a stationary inflection point. If the first nonzero derivative is even (fourth, sixth etc.), the point is minimum if that derivative is positive and maximum if negative.

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Got it!snorkack said:

In that case, if the first nonzero derivative is odd (third, fifth etc.), it is neither maximum nor minimum but a stationary inflection point. If the first nonzero derivative is even (fourth, sixth etc.), the point is minimum if that derivative is positive and maximum if negative.

That seems quite counter-intuitive... is there a way to demonstrate it "empirically", or is there just an analytical way to do it?

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