#stationary-points

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mediumAI-checkedOptimization & Calculus

A smooth function has $f'(0) = 0$ and $f''(0) = 0$ . What can you conclude about the point $x = 0$ ?

It is not a stationary point, because the second derivative vanishes1
Nothing yet: 0 may be a minimum, a maximum or neither2
It is a horizontal point of inflection, as for $f(x) = x^3$ at 03
It is a local minimum, because the function is flat to second order4

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