182 
CALCULUS 
If, in particular, 
(13) F(x, y, z) = Ax 2 + By 2 + Cz 2 , 
the equation (12) reduces to 
(A + A)(B 4- A)((7 -(— A) = 0, 
and the roots, which are all real, are — A, — B, and — C. 
In the general case, equation (12) has at least one real root, and 
no root of (12) is 0. For, the vanishing of the determinant arising 
by setting X = 0 in (12) would mean that F(x, y, z) could vanish for 
values of x, y, z not all 0. 
Consider the function F(x, y, z) in the points of the sphere 
$ = a 2 . Since F(x, y, z) is continuous, it has a maximum value on 
the sphere, and also a minimum value.* Hence two of the three 
roots of (12) are real and distinct, and thus all three are real. 
Let the coordinate axes be so rotated that F attains its maximum 
value in the point (0, 0, £), where £ > 0. From (11) it appears that 
/1=0, E = 0, X — — C. Thus 
F(x, y, z) — Ax 2 2 Fxy + By 2 + Cz 2 , C > 0. 
If F 0, a suitable rotation of the axes about the axis of z will 
remove the term in xy, and thus the form (13) is attained, where A, 
B, C are all positive. 
EXERCISES 
1. Find the values of (x, y, z) for which the function 
u = xyz 
is stationary, if x + y + z = 1. 
2. Work Example 2 of §3 by means of Lagrange’s multipliers. 
3. Examine the Exercises at the close of §3 and determine to 
which of these the method of Lagrange’s multipliers is particularly 
well adapted. 
4. Show that the method of Lagrange’s multipliers is valid in the 
case of functions of a single variable, given in the form: 
u = F(x, y) <!>(;», y)— 0. 
* If, in particular, these values are the same, i.e. if F(x, y, z) is constant on 
the sphere, then F(x, y, z) = K<b(x, y, z) (K = const.) and all three roots of 
(12) are real and equal.