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

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

• A. It is a local maximum
• B. It is an inflection point
• C. It is a saddle point
• D. It is a local minimum

Answer: (D)

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.

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.