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Can third order derivative (jerk) be use for optimization?

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In optimization, ussually people just use second order derivative (or Hessian matrix) to classified local extrema. Suppose that my function is 3 times differentiable, what can I use this information for ?

As a follow up I really want to ask about the analog of the jerk (third order derivative) in case of multivariable optimization. Is it some sort of higher dimension matrix ?

All of my questions are just coming from pure curiosity.

Please help me with this

Thank you for your enthusiasm !


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