5.9 STATIONARY VALUES UNDER CONSTRAINTS
we findz=− 2 x. Substituting these values into the first constraint,x^2 +y^2 +z^2 =1,we
obtain
x=±1
√
6
,y=±1
√
6
,z=∓2
√
6
. (5.38)
Because of the high degree of symmetry amongst the equations (5.34)–(5.36), we may obtain
by inspection two further relations analogous to (5.37), one containing the variablesy, z
and the other the variablesx, z. Assumingy=zin the first relation andx=zin the
second, we find the stationary points
x=±1
√
6
,y=∓2
√
6
,z=±1
√
6
(5.39)
and
x=∓2
√
6
,y=±1
√
6
,z=±1
√
6
. (5.40)
We note that in finding the stationary points (5.38)–(5.40) we did not need to evaluate the
Lagrange multipliersλandμexplicitly. This is not always the case, however, and in some
problems it may be simpler to begin by finding the values of these multipliers.
Returning to (5.37) we must now consider the case wherex=y; then we find
3(x+y)+2λ=0. (5.41)However, in obtaining the stationary points (5.39), (5.40), we didnotassumex=ybut
only requiredy=zandx=zrespectively. It is clear thatx=yat these stationary points,
and it can be shown that they do indeed satisfy (5.41). Similarly, several stationary points
for whichx=zory=zhave already been found.
Thus we need to consider further only two cases,x=y=z,andx,yandzare all
different. The first is clearly prohibited by the constraintx+y+z= 0. For the second
case, (5.41) must be satisfied, together with the analogous equations containingy, zand
x, zrespectively, i.e.
3(x+y)+2λ=0,3(y+z)+2λ=0,3(x+z)+2λ=0.Adding these three equations together and using the constraintx+y+z= 0 we findλ=0.
However, forλ= 0 the equations are inconsistent for non-zerox,yandz. Therefore all
the stationary points have already been found and are given by (5.38)–(5.40).
The method may be extended to functions of any numbernof variablessubject to any smaller numbermof constraints. This means that effectively there
aren−mindependent variables and, as mentioned above, we could solve by
substitution and then by the methods of the previous section. However, for large
nthis becomes cumbersome and the use of Lagrange undetermined multipliers is
a useful simplification.