Vector Calculus · Investigation

Lagrange Multipliers

Why do constrained extrema occur where gradients become parallel?

Follow a point around a level curve of $g(x,y)$ . Its projection remains on the $xy$ -plane, while the lifted curve traces the height of $f(x,y)$ . Where does the height stop increasing or decreasing?

One geometric workspace

Drag the silver point around the constraint curve, or orbit the 3D scene

z_0

9.00

● Projection● Curve on f● Curve on g● ∇f (plane)● ∇g (plane)

Current point

(2.77, 1.15)

f(x,y) = 3.18

Rate along the curve

6.36

d/dθ f(x(θ),y(θ))

Gradient alignment

0.71

|sin(angle between gradients)| · 0 means aligned. Move the point to find an extremum.

Why do the gradients align?

Motion along the constraint is tangent to its level curve, so it is perpendicular to $\nabla g$ . At a constrained local maximum or minimum, the derivative of $f$ in that tangent direction is zero. Provided the constraint is regular (its gradient is nonzero), the two gradients must therefore be parallel.