Does $f(x) = x^3$ have a local extremum at $x = 0$ ?

No:

f'

does not change sign at 0, so it is a point of inflectionCorrect answerYour answerYes, a local minimum, because

f'(0) = 0

Correct answerYour answerYes, a local maximum, because

f''(0) = 0

Correct answerYour answerYes, both a maximum and a minimumCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$f'(x) = 3x^2$ is positive on both sides of 0.
$f'(0) = 0$ makes 0 a critical point, but with no change of sign there is no maximum or minimum.

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Does f(x)=x3f(x) = x^3f(x)=x3 have a local extremum at x=0x = 0x=0?