So grad(f) must be perpendicular, otherwise if it has components along the surface you just follow them until you reach a point that doesn't budge.įinally, all grad(g) are normal to g since they are level curves to g = 0. Since gradients point in directions of max increase, a max or min should not point in any directions along the curve/surface, because if they do clearly there is something larger/smaller immediately next to it. To see why extrema of f must have gradients that are perpendicular to the curve/surface g, consider a non-perpedicular grad(f) - it can't possibly be a max or min because it has components pointing in the directions of the surface. The reason is that when f(x,y) is constrained to the curve/surface g(x,y), we need to find a point (a,b) such that grad(f(a,b)) is perpendicular to this curve/surface g(x,y) which is a candidate for an extrema.

You have to say why the gradient of f is a multiple of gradient g. I don't feel this explains the essence of Lagrange multipliers.

Optimization calculus khan academy