Three Graphs, One Function, Three Questions

You'll learnto sort any fact about f, f′ or f″ by asking which curve it belongs to, and whether it is a value, a sign, or a change of sign.

One function, drawn three times on the same x axis, with a rule that cuts all three at once. The same x gives a height on one curve, a direction on the next, and a bend on the third. Everything in this topic comes down to asking which curve a fact belongs to, and whether that fact is a value, a sign, or a change of sign. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · A ROOT IS A CANDIDATE, A SIGN CHANGE IS THE VERDICT · v2

f(x) = 3x⁵ − 20x³ · a value on f, a sign on f′, a sign on f″ · a change of sign is the verdict

Before you start

What you're looking at

One function, f(x) = 3x⁵ − 20x³, drawn three times on a shared x axis: itself, its first derivative, its second derivative.

The question

Given a fact about one of the three curves, what does it let you say about f?

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