Three Graphs, One Function, and a Different Question for Each

You'll learnto sort a pile of facts about f, f′ and f″ by deciding which function each one is about — and to tell a value apart from a sign, and a sign apart from a change of sign.

Nothing new is introduced in this topic. What is new is that everything arrives at once — a table, or a single graph — and has to be sorted before it can be used. Two questions settle every item. Which function is this fact about: f, f′ or f″? And does the conclusion need a value, a sign, or a change of sign? A value tells you where to look. A sign tells you what is happening. Only a change of sign licenses a verdict, which is why a root of f′ is a candidate rather than a maximum, and a root of f″ is a candidate rather than an inflection point.

8 STEPS · 6 QUICK CHECKS · A ROOT IS A CANDIDATE, A SIGN CHANGE IS THE VERDICT · v1

f(x) = 3x⁵ − 20x³ ⇒ f′(x) = 15x²(x² − 4) ⇒ f″(x) = 60x(x² − 2)

Before you start

What you're looking at

One function, f(x) = 3x⁵ − 20x³, drawn three times: itself on top, its first derivative in the middle, its second derivative underneath. All three panels share the same x axis, so a vertical line cuts all three at the same input.

The question

Given a fact — a number, a sign, a crossing — which of the three functions is it about, and what does it license you to conclude about f?

Watch for

The one x where f′ touches zero and comes straight back without changing sign, and the difference between that and the three places where f″ genuinely crosses.

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