Mistake Master · AP Calculus · Unit 5 · Step-Through Animation
One Number Settles It, Unless That Number Is Zero
You'll learnto classify a critical point from a single value of f″, and to recognise the two situations in which this test declines to answer — so that a refusal never gets written down as though it were a result.
Topic 5.4 classified a critical point by reading the sign of the first derivative on both sides of it. This test does the same job by reading one number instead: the second derivative, at the point. A horizontal tangent under a dome is the top of the dome; a horizontal tangent inside a bowl is the bottom of the bowl. That is the whole idea. What you buy the speed with is coverage — the test fails outright when the second derivative comes out zero, and it never starts at all when the first derivative was not zero to begin with. Knowing when it has failed is most of the skill, because a refusal looks exactly like data.
8 STEPS · 6 QUICK CHECKS · f″(c) = 0 IS A REFUSAL, NOT A VERDICT · v1
A cubic, f(x) = x³ − 3x, with a flat tangent at each of its two critical points, and the graph shaded by which way it bends: f″ < 0 on the left, f″ > 0 on the right.
The question
At a point where the tangent is horizontal, what does one value of f″ tell you — and what does it tell you when that value comes out zero?
Watch for
The two ways this test hands back nothing: a second derivative of zero, and a critical point where the first derivative has no value at all. Neither of those is a verdict.
Mistake Master