Second Derivative Test D = AC − B²

Surface: z = ½(Ax² + 2Bxy + Cy²) — the local shape of any function near a critical point

Bending team A·C

2.25

Twist penalty B²

0.00

D = AC − B² = 2.25

LOCAL MINIMUM — a bowlBending wins, and A > 0 (happy face)

Probe direction θ k(0°) = 1.50

k(θ) = A·cos²θ + 2B·sinθcosθ + C·sin²θ — curvature of the slice in direction θ

x-slice (curvature A) y-slice (curvature C)
probe curving up probe curving down
Drag to orbit · scroll to zoom

Bending (A·C) wants a clean bowl or dome — x and y curving the same way. Twisting (B²) tries to fold it into a Pringles chip. Press ▶ Play the twist battle to watch B grow until the twist tears the bowl into a saddle.