Lagrange multipliers are used in optimising or maximising a function
subject to one or more equality constraints
The solution is found by minimizing
with respect to the
and
The Lagrange multipliers are the
For example to maximise
subject to
form the expression
Differentiating with respect to
gives
(1)
Differentiating with respect to
gives
(2)
Differentiating with respect to
gives
(3)
From (1)
From (2)
From (3)
The maximum value of
is then
This is illustrated below. The maximum is where the curves touch.
