LaGrange Multiplier

From 3D Printables

When looking for the extreme values of a differentiable function f, subject to two constraints, g=0 and h=0, often the gradients of $\del g$ and ∇h are not parallel. In such cases ∇f is a linear combination of ∇g and ∇h. It is often difficult for student to visualize the two intersecting surface, the tangent to the curve of intersection, the gradients which at perpendicular to the surfaces, and the gradient of f. This model helps students see by examining the three dimensional model, that ∇f lies in the plane determined by the gradients to the level sets, and that that plane is perpendicular to the tangent to curve determined by the intersection of the surfaces g=0 and h=0. Hence it is more readily apparent why the optimal solution should satisfy ∇f= λ∇g + μ∇h and g(x,y,z)=0, h(x,y,z)=0. Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha\,