Lagrange Multipliers




     Sometimes  we  need to find the extreme values of a function
whose domain is constrained to lie within some particular  subset
of the plane.

Example:  we  want  to find the points on the hyperbolic cylinder
xˆ2 ‐ zˆ2 ‐ 1 = 0 that are cloest to the origin.



That is, we want to minimize f(x,y,z) = xˆ2 + yˆ2 + zˆ2

We can treat x, y as the independent variables, or better,  treat
y,  z  as the independent variables. Then by the First Derivative
Test, find x, y and z.


Another solution:



At each point of contact, the cylinder and sphere have  the  same
tangent plane and normal line.

Then we have f(x,y,z)=xˆ2+yˆ2+zˆ2 ‐ aˆ2 = 0 and

g(x,y,z) = xˆ2 ‐ zˆ2 ‐ 1 = 0

At any point of contact, the gradients f and g will be parallel:



No  point on the surface has a zero x‐coordinate to conclude that
x != 0.

To satisfy this, z = 0, y = 0, λ=1, x=+‐1

This is the method of Lagrange multipliers. The method says  that
the  local  extreme values of a function f(x,y,z) whose variables
are subject to a constraint g(x,y,z)=0 are to  be  found  on  the
surface g = 0 among the points where




Another Example:










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