MAT37x

Franklin College

Spring2004

Erich Prisner,

## Optimizing on surfaces with restraints:

# Lagrange Multiplier

Let
f be a function
from **R**^{2}
into **R**.
Remember how to find global maximum points
for this surface. What if we are restricted to a certain region? The boundary of
the region should be given by one equation g_{i}(x,y)
= h.

How do we find the maximum f-value on this g-path?
The idea is simple. Assume we are at point on this boundary path
where the gradient vector is *not* perpendicular to the path. Then we could gain
f-value by moving along. Therefore, at the highest
f on the boundary, the gradients of
f and g should point
in the same direction, i.e.

(f_{x},f_{y}) = l
(g_{x},g_{y}) .

In this way we get two equations with three unknowns, namely
x, y, l. The third equation
we need to solve the system is g(x,y)=h.

### Example: