AP Calculus AB and BC

First vs Second Derivative Test

Q: If the second derivative test is inconclusive, is there no extremum?

No. Inconclusive means you need a different test. Fall back on the first derivative test, which always gives an answer.

The first derivative test classifies a critical point by whether the derivative changes sign there, and it never fails. The second derivative test checks the sign of the second derivative at the point, which is faster but says nothing when that value is zero.

First derivative test

Use when: You want a method that always reaches a conclusion, or the derivative does not exist at the point.

Second derivative test

Use when: The second derivative is easy to compute and you only need a quick classification.

Side by side

	First derivative test	Second derivative test
Uses	Sign of $f'$ on both sides	Sign of $f''$ at the point
Maximum when	$f'$ goes positive to negative	$f''(c) < 0$
Can be inconclusive	No	Yes, when $f''(c) = 0$
Works where $f'$ is undefined	Yes	No

The second derivative test is really a statement about concavity. At a critical point on a concave down curve the graph must turn over, giving a maximum. When $f''(c) = 0$ the curve is not committing to a direction, so the test has nothing to say.

Compare $x^4$ , $-x^4$ , and $x^3$ at the origin. All three have $f'(0) = f''(0) = 0$ , yet they have a minimum, a maximum, and neither. Only the first derivative test separates them.

For justification credit

On free response, the credited justification for the first derivative test names the sign change of $f'$ . Writing that $f'$ changes from positive to negative at $x = 2$ is what earns the point.

Frequently asked questions

If the second derivative test is inconclusive, is there no extremum?

No. Inconclusive means you need a different test. Fall back on the first derivative test, which always gives an answer.

In the CED: Unit 5: Analytical Applications