Question

why f(x) = x^3 doesn't have relative extrema at 0

Answer 1

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Answer 2

Its just the nature of the function, it is monotonically increasing without bound.

Answer 3

Even though the derivative of the function, f'(x) = 3x^2 equals to 0 at x = 0, it does not change signs before x = 0 or after it; it always remains positive. For a function to have a local extrema at a point, its derivative must equal to 0 and change signs there.