Lagrange as a kiss

Want: f = x² + y² as small as possible (circles). Must: stay on the line x + 2y = c. The light circle is the contour through the dot; the arrows are ∇f and ∇g there.

f at the dot

2.240

min on the line, c²/5

0.800

angle ∇f to ∇g

53.3°

λ = 2c/5 = d(min)/dc

0.800

The contour through the dot crosses the line, so sliding one way along the line reaches a smaller circle. Not done yet.