Peaks, Pits & Passes

The second derivative test

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Critical points have zero gradient: maxima, minima or saddles.
• $D = f_{xx}f_{yy} - f_{xy}^2$: positive means max or min (by the sign of $f_{xx}$), negative means saddle.
• Contours are closed loops around extrema and cross like an X at saddles.

Steps

1. Flat isn’t enough
2. A bowl
3. From bowl to saddle
4. The simplest saddle
5. Two critical points
6. Classify it