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Case 3 A similar approach applies to one functions, f,
with three
independent variables, x, y, and z subject to two constraints.
The result is 5 equations in the following 5
unknowns:
x, y,
z, λ1 and
λ2
Here λ1 and
λ2
are two Lagrange multipliers
one for each constraint.
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The function is:
f(x,y,z) =
0,
The constraints are: g(x,y,z) =
0, and h(x,y,z) =
0
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For a maximum or for a minimum:
∂f/∂x =
λ1 [
∂g/∂x ] + λ2 [ ∂h/∂x ]
∂f/∂y =
λ1 [
∂g/∂y ] + λ2 [ ∂h/∂y ]
∂f/∂z =
λ1 [
∂g/∂z ] + λ2 [ ∂h/∂z ]
with g(x, y,
z) = 0
and h(x, y,
z) = 0
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Once
one solves this system of equations for x, y, z,
λ1 and λ2 ,
then
the values of (x, y,
z) can be input into f(x, y,
z) to obtain the optimal value.
Click
here for examples.
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