AP Calculus BC Practice Test

If f''(a) > 0 and f'(a) = 0, what does this say about the point a on the graph?

It is a local maximum It is an inflection point It is a saddle point It is a local minimum

Correct Answer:

It is a local minimum

Explanation:

This uses the second derivative test for a critical point. If f'(a) = 0, a is a candidate for a local extremum. When f''(a) > 0, the graph is concave up at a, like a bowl, so values of f nearby are greater than f(a) and the tangent is horizontal there. That makes a a local minimum. If f''(a) < 0, it would be a local maximum, and if f''(a) = 0 the test is inconclusive. An inflection point would require a change in concavity, which does not occur here since the second derivative is positive. So the point a is a local minimum.

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