Reading Shapes

What f′ says about f

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $f' > 0$: rising. $\;f' < 0$: falling. $\;f' = 0$: a critical point.
• If the sign of $f'$ goes $+ \to -$ it's a maximum. If it goes $- \to +$ it's a minimum.
• Second derivative test: $f'' < 0$ at a critical point means a maximum, $f'' > 0$ a minimum.

Steps

1. Uphill
2. Match the derivative
3. Critical points
4. Find the valley
5. Critical points of a cubic
6. Steepest on the way down
7. The second derivative