calculus and analysis - How can I implement the method of Lagrange multipliers to find constrained extrema?

I want to form the function $h=f-\lambda_{1}g_{1}-\lambda_{2}g_{2}$ where $f$ is the function to optimize subject to the constraints $g_{1}=0$ and $g_{2}=0$ so that I can solve the first partial derivatives with respect to $\lambda_{1}$ and $\lambda_{2}$. Can someone get me started using $f(x,y,z)=xy+yz$ subject to the constraints $x^2+y^2-2=0$ and $x^2+z^2-2=0$?

Answer

We define the function f and multiple constraint functions g1, g2:

f[x_, y_, z_] := x y + y z
g1[x_, y_] := x^2 + y^2 - 2
g2[x_, z_] := x^2 + z^2 - 2

then, in order to find necessary conditions for constrained extrema we introduce the Lagrange function h with Lagrange multipliers λ1 and λ2:

h[x_, y_, z_, λ1_, λ2_] := f[x, y, z] - λ1 g1[x, y] - λ2 g2[x, z]

Now we solve an appropriate system of equations satisfying necesary conditions (i.e. vanishing of all first derivatives of h):

TraditionalForm[ 
  Column[ pts = {x, y, z} /. 
          FullSimplify @ Solve[ D[h[x, y, z, λ1, λ2], #] == 0 & /@ {x, y, z, λ1, λ2}, 
                                      {x, y, z, λ1, λ2}], Frame -> All]]

A bit nicer way of finding all the solutions uses Grad - a new function in Mathematica 9 for vector analysis:

{x, y, z} /. Solve[ Grad[ h @@ #, #] == 0, #]& @ {x, y, z, λ1, λ2} // FullSimplify

The above table contains all critical points of the Lagrange function h. For sufficient conditions one can use Maximize and Minimize, e.g.:

FullSimplify @  ToRadicals @ 
Maximize[{f[x, y, z], g1[x, y] == 0, g2[x, z] == 0}, {x, y, z}]

{1 + Sqrt[2], {x -> -(1/Sqrt[2 + Sqrt[2]]), 

               y -> -Sqrt[1 + 1/Sqrt[2]],
               z -> -Sqrt[1 + 1/Sqrt[2]]}}

We add a graphics with contours of constrained minima and maxima, the contraint functions ass well as all critical points of h:

Show[ 
  ContourPlot3D[{ f[x, y, z] ==  1 + Sqrt[2], 
                  f[x, y, z] == -1 - Sqrt[2], 
                  g1[x, y] == 0, g2[x, z] == 0}, 
                  {x, -2.3, 2.3}, {y, -2.3, 2.3}, {z, -2.3, 2.3}, 
                  ContourStyle -> {Directive[Cyan, Opacity[0.5]], 

                                   Directive[Green, Opacity[0.5]], 
                                   Directive[Orange, Opacity[0.15]], 
                                   Directive[Orange, Opacity[0.15]]}, Mesh -> None], 
  Graphics3D[{Magenta, PointSize[0.015], Point[pts]}]]

On the cyan surfaces we have maxima, on the green ones - minima and the solutions of the necessary conditions are denoted with the magenta points lying on the tube constraints.