A Falling Curve Can Bend Upward

You'll learnto read which way a graph bends from the sign of f″, and to keep that apart from which way it is going.

Going up or down is one question, and it belongs to f′. Bending is a different question, and it belongs to f″. A curve can fall while it bends upward, and most falling curves do. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · CONCAVE UP MEANS f′ IS INCREASING · POSITION IS DIRECTION, SHAPE IS CONCAVITY · v2

f(x) = x³ − 3x² − 9x + 5 ⇒ f′(x) = 3(x + 1)(x − 3) ⇒ f″(x) = 6x − 6

Before you start

What you're looking at

One cubic, f(x) = x³ − 3x² − 9x + 5, and later four other curves. Stacked panels share one x axis, so a vertical line cuts them at the same input.

The question

Which way is the graph bending? Not which way it is going: that is a different question with a different answer.

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