At x = 0, the function f(x) = x³ has:

Model Answer & Options

Source: Extra Practice

A local maximum

A local minimum

A point of inflection

Discontinuity

Explanation

f'(x) = 3x², so f'(0) = 0. However, f''(x) = 6x, so f''(0) = 0, and the second derivative test is inconclusive. Using the first derivative test, f'(x) = 3x² is positive for x 0. Since the sign of the derivative does not change, there is no extremum. Because f''(x) changes sign at x=0 (from negative for x0), it is a point of inflection.

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