Chapter 3: Many Variables; Section 1: Differentiation; page 6
Constraints, Exercise 1
Question
Use Lagrange multipliers to find what is the shortest distance of the plane
from the coordinate's origin! (suggestion: first calculate the Lagrange multiplier.)
Reminder
It can be shown that the method of Lagrange multipliers can be applied to obtain the stationary points of a function of any number n of variables, subject to any number of constraints k, providing that k<n. For this purpose one has to define k Lagrange multipliers, one for each constraint.
(3.1.6.32)
For calculating the stationary points of the function, one has to use the k constraints of (3.1.6.32), in addition to the following n equations:
(3.1.6.33)
Parts 1-7
The distance squared that will be minimized is
with the constraint of the plane:
According to (3.1.6.32-33), the equations to be solved are
In this case one can easily obtain the solution of λ, by multiplying each equation by an appropriate factor, written on the right, and by adding up the resulting equations:
The result of this procedure is the equation
or
The substitution of λ obtained in part 4, in the first equation of part 3, yields
similarly
and
yielding
and the required minimal distance is
Score
Note: The Lagrange multiplier λ does not have to be calculated first, or even calculated at all. Any other correct ways to solve the equations of part 2 are legal, but more complicated.