Lagrange Multipliers

Why Parallel Gradient Vectors?

We wish to optimize a point on the surface

f(x,y) = xy

given the constraint

g(x,y) = x^2+y^2 = z_0

for a given height level $z = z_0.$ You can toggle between the surfaces ( $f$ and $g$ ) below. The constraint curve is shown below that with a movable point $(x,y)$ on the constraint. The point and constraint curve are lifted to both surfaces. Check the boxes in the constraint diagram to see the gradient fields of $f$ and $g$ . Note that movement on the constraint is always orthogonal to the gradient of $g$ (why?). What happens when that movement is also orthogonal to the gradient of $f$ ? The highest and lowest points on the blue surface occur at points where the gradient vectors align. Can you see it? Why do you think that happens?

Surfaces

Optimization Surface: f(x,y)

Level Set Surface: g(x,y)

g(x,y)=z_0

z_0

25.0