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CHAPTER XVI.

MAXIMA AND MINIMA VALUES OF A FUNCTION OF SEVERAL

VARIABLES.

236. LET u = (x, y, z) be a function of three independent variables, of which we require the maxima and minima values. By an investigation similar to that in Art. 224,

$(x+h, y + k, z + 1) − 4 (x, y, z)

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where R is a function involving powers and products of h, k, of the third degree, which may be expressed for abbreviation by

v denoting

1 d
h

d
+k
+7

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dz

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3 dx dy

(x+Oh, y + Ok, z+Ol).

If we make h, k, 7 small enough, the sign of

$ (x+h, y + k, z + 1) − 4 (x, y, z)

will depend upon that of the terms involving only the first powers of h, k, l; hence, to ensure a maximum or minimum,

we must have

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and therefore, since h, k, l are independent,

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Let values of x, y, z be found from these equations, and

when these values are substituted in

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d'u

d'u

&c., let

dx

dy

27

=B, C,

A', dx dz

dz2

=

=

d'u B', dx dy

=

C'.

$ (x + h, y + k, z + 1) − p (x, y, z)

can, with the values of x, y, z just found, be made to depend on that of

Ah2 + Bk2 + Cl2+2A′kl + 2B′hl + 2 C'hk ......... (1). Hence, that u may have a maximum or minimum value, the expression (1) must retain the same sign, whatever be the signs and values of h, k, l comprised between zero and fixed finite limits. If we put

it follows that

h=sl, k=tl,

As2+B+C+2A't + 2B's + 2 C'st............(2),

must be of invariable sign, whatever be the signs and values of s and t. Multiply (2) by A, and rearrange the terms; then (As + B' +'C't)2 + (AB − C12) ť2 + 2 (AA' – B'C')t +AC – B22

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Hence, (ABC'2) ť2 + 2 (AA' – B'C')t + AC — B" must be incapable of becoming negative; therefore

AB-C" must be positive, and .......

(4),

(AA' – B'C') less than (AB- C'2) (AC – B12)................(5) ;

(4) and (5) are the conditions that must be satisfied in order that u may be a maximum or minimum. Conversely, if they

are satisfied, u is a maximum or minimum; for then (3) is necessarily positive, therefore (2) has always the same sign as A, and u is a maximum if A be negative, and a minimum if A be positive.

Hence the necessary and sufficient conditions for the existence of a maximum or minimum value of a function u of three independent variables, are, that the values of x, y, z drawn from

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(d2u d'u

d2u

{de dy-(drdy)}{de do- (drde)"}·

dz2 dx dz

It follows of course from these conditions, that

and thus

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dx2 dz2 dx dz

d'u d'u d2u

dx2, dy2' dz2

must all have the same sign, and u

is a maximum if that sign be negative, and a minimum if it be positive.

From the conditions (4) and (5), we should conjecture by the principle of symmetry, that BC-A" will also be positive if (4) and (5) hold. This is easily verified, for from (5) we find that

A{ABC+2A'B'C' - AA" - BB" - CO"} – – is positive, and therefore, since by (4) A and B have the same sign,

B{ABC+2A'B'C' — AA" — BB” — CC12}

is positive, and therefore

(BB' — A'C')2 is less than (BC- A”) (BA — C′′2),

from which it follows that BC-A" is positive.

237. Ex. Let u =

du

=

xyz

(a + x) (x + y) (y + z) (z + b) '

yz (ay — x2)

=

u (ay - x2)

dx ̄ ̄ (a+x)2 (x + y)2 (y + z) (≈z + b) x (α + x) (x + y)

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Hence, if ay-x2 = 0, xz — y2 = 0, and by-z2=0, u may be a maximum or minimum: these equations give

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Proceeding to the second differential coefficients of u, we

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the terms included in the &c. being such as vanish when the specific values are assigned to x, y, z.

Hence A

2u

==

2

a3r (1+r)2 a3r (1+r)* *

Similarly B, C, &c. can be found, and we shall finally arrive at the result that u is a maximum.

238. Suppose it required to determine the maxima and minima values of a function (x, y, z,...) of m variables, these variables being connected by n equations, of which the general form is

....

F,(x, y, z, ...) = 0................................ .(1). The m variables involved in & are of course not all independent, since by means of the given equations n of them

may be expressed in terms of the remaining m-n. The simplest theoretical method of investigating the maxima and minima values of would be to express by means of the given equations the values of n of the variables in terms of the rest, and to substitute these values in ; thus would become a function of m-n independent variables, and we might proceed to ascertain its maxima and minima values in the manner already given for functions of one, two, or three independent variables. But this method would be often impracticable on account of the difficulty of solving the given equations, and the following is therefore adopted.

Suppose x, y, z... all functions of some new variable t, of which consequently becomes a function. Put for shortness

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dy Dx, dt

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From the n given equations (1) we deduce

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By solving the linear equations (3) we can express n of the quantities Dx, Dy, Dz... in terms of the remaining m n. Substitute these values in (2), then only m―n of the quantities Dx, Dy, Dz... remain, and we have a result which may be written

do

dt

=X.Dx + Y. Dy + Z.Dz...+Q.Dq.

..

(4),

where X, Y, Z, ... do not involve any of the quantities Dx, Dy, Dz, &c. Since, consistently with the given _equations, we may consider the m-n quantities Dx, Dy, Dz, ...

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