MAXIMA AND MINIMA 
175 
Auxiliary Variables. As in the case of functions of a single varia 
ble, so here it frequently happens that it is best to express the 
quantity to be made a maximum or a minimum in terms of more 
variables than are necessary, one or more relations existing between 
these variables. The student must, therefore, in all cases begin by 
considering how many independent variables there are, and then write 
down all the relations between the letters that enter; and he must 
make up his mind as to what letters he will take as independent 
variables before he begins to differentiate. 
Example 2. What is the volume of the greatest rectangular paral 
lelepiped that can be inscribed in the ellipsoid: 
(2) 
+ y+ 
b 2 
1? 
We assume that the faces are to be parallel to the coordinate 
planes and thus obtain for the volume: 
V = 8 xyz. 
But x, y, z cannot all be chosen at pleasure. They are connected by 
the relation (2). So the number of independent variables is here 
two, and we may take them as x and y. We have, then: 
(3) 
cV 
dx 
= 8 y[z + »v]= 0, 
dx 
From (2) we obtain : 
(4) 
GZ 
dx 
C L X 
dV q f , dz, n 
= 8x1 z + y — )= 0. 
oy V °Vj 
dz c 2 y 
By b 2 z 
Now, neither x = 0 nor y — 0 can lead to a solution of the problem, 
and hence it follows from (3) and (4) that 
n2 o*2 _ c 2 ?/ 2 A 
z - V = 0 ’ 
c L x L n 
z — = 0, 
a 2 z 
b 2 z 
V _ z 
b 2 c 2 
Thus the parallelepiped whose vertices lie at the intersections 
of these lines with the ellipsoid, i.e. on the diagonals of the cir 
cumscribed parallelepiped x = ± a, y = ± b, z = ± c, is the one 
required,* and its volume is 
F= f V3 abc. 
* The reasoning, given at length, is as follows. V is a continuous positive 
function of x and y at all points within the ellipse 
* 2 , V 2 _ 
(j! i) J 
1,