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I. F. Quinby Horatio Nelson Robinson.

# A new treatise on the elements of the differential and integral calculus

. (page 10 of 22)

II. Of all the squares inscribed in a given square, which
is the least ?

Ans. That having the vertices of its angles at the middle
of the sides of the given square.

12. Inscribe the greatest rectangle in a given semi-ellipse.
Let the equation of the ellipse be

a^y'^ + b^x^=ia^b\

b

j The sides a y^2,â€” ^, and its
( area is ab.

13. Given the whole surface of a cylinder, required its
form when its volume is a maximum.
Represent the whole surface by 2;ra^.

Ans.

Eadius of the base â€”^, axis -â€” ^,

volume â€” r- .
L 3t

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SECTION XII.

EXPANSION OP FUNCTIONS OF TWO OR MORE INDEPENDENT VARI-
ABLES, AND INVESTIGATION OP THE MAXIMA AND MINIMA OF
SUCH FUNCTIONS.

121* Let it be required to develop, and arrange according
to the ascending powers of A and h, the function F{x-\-h, y + A:),
when F{x, y), and all its partial derivatives up to those of the
n*** order inclusive, are finite and continuous for all values of a?
and y included between the values x and x-^hj y and y -\-k;
h and k themselves being finite.

For the time, replace h and k by ht and kt respectively ; so
that, when it is desired, we may pass back to h and k by mak-
ing ^ = 1. Then F{x + A, y + k) becomes F{x + lit, y + kt).
Now, by hypothesis, x^ y. A, k, and t, are all independent of
each other; and, considered with reference to t alone, we may

write

F{x + ht,y + kt)=f{t) (1),

F{x,y)=f{0) (2).

For all values of t between the limits ^ = and ^ = 1, it is
evident that /{t) and its derivatives, up to those of the n^^ or-
der inclusive, satisfy the conditions above imposed on F(x, y)
and its derivatives.

Hence, for such values, we have, by Maclaurin's Theorem,

m =/(o) +/'(o) \ +/"(o) i4 + â€¢ â€¢ â€¢

188

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EXPANSION OF FUNCTIONS, 189

Deducing the values of /(O), /(O), /'(O) . . ./"HO. by the
method pursued in Art. 110, except that now x and y are not
replaced by the particular values a and h, and substituting
them and that of/(<) in Eq. 3, we have

F{x + ht,y-^U) = F{x,y) + \(^f-h + f-l^

t^ /d^F d^F d^F d^F \

+ iMd^''+'d^/''+'d^^'''+d^'')
+

+ l.:i...nU^" ^""d^^^^d'y * + - +^>-+Â»Â« (4).

The notations a; = ir + ^A/, y = y-^ dU, attached to the pa-
renthesis of the last term, signify that in the derivatives

d^F d^F d^F . , . ^ . .7. . ,.

cS^' dx^^^=^dy ' â€¢ â€¢ â€¢ d^' ^ ^^ replaced by re + dht, and y by

y + dht

In (4), make ^ = 1, and we have

1 /rfÂ«i^, d"i?^ ^ ,, d^'F \

which is the development sought.

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190 DIFFERENTIAL CALCULUS.

If, in Eq. 5, we make a; = and y = 0, and then in the re-
sult write X for A, and y for &, we find

/dF\ fdF\

y=0 y=0

y =

â– -ev y = ep

which is the formula for the development of a function of two
independent variables into a series arranged according to the
ascending powers of the variables.

The extension of formulas (5) and (6) of this article to func-
tions of more than two variables is easily made. For the
expansion of F{x + h, y +k, z + i...), we should find
F{x + h, y + k, z + i...) = F{x,y,z...),
dF^ dF, . dF. ,

^ 1.2 \dx' ^ dy' ^ ^ dxdy ^ J

^ l.2.6\dx^ ^ ^ dx^dy dxdy^ /

+

1 /d'F , , d-F^^ ,

^ 1.2... n Kdx" ^dy"

"^"F ^ ,, , \

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EXPANSION OF FUNCTIONS. 191

and if, in this, we first make a:, y, 2 . . . , severally equal to zero,
and then in the result write a;, y, Â« . . ., for A, i, i . . ., respec-
tively, we have

(rfTP\ /f7J^\ /{97^\

+m|Q"+-+(^,h+

+ 3 T-.-iWy' +

+

1 f/(;Â»J?'\ . /d''F\ ,

+ i:2::rÂ«|(^>"+(^>"+-

a formula for the development of a function of any number of
independent variables, and in which the notation ( )q signifies
that the variables entering the expression within the paren-
theses are made zero. In formulas from (5) to (8) inclusive, of
ference between the value of the sum of the preceding terms
of the development and the value of the function. When the
form of the function under consideration, and the values at-
tributed to the variables, are such, that, aan increases without

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192 DIFFERENTIAL CALCULUS.

limit, the remainder decreases without limit, then, by taking n
sufficiently great, the remainders may be neglected.

122. Maxima and minima of functions of two or more in-
dependent variables.

A function F{x,y, z...) of several independent variables
is a maximum, when, being real, it is, for certain values of
the variables, greater than F{x-\-h, y-\-Jc, 2 + z...); the
increments /i, kji.. ,, being very small, and taken with all pos-
sible combinations of signs. On the contrary, the function is
a minimum, when, under the same conditions, it is less than
F{x-\-hj y-\-h, z + i...). Let us consider first the func-
tion F{x, y) of the two independent variable.s x and y, and
endeavor to deduce from the conditions of these definitions,
the criteria of a maximum or minimum of tliis function.
Resuming Eq. 5, Art. 121, stopping in the second member at
those terms which involve the third order of the partial deriv-
atives of the function, and transposing F{x^ y) to the first
member, we have

1 fd'F^^ ^cf-F ^^ d'-F \

1.2.3 Vc/x^ ' dx'dy ' dxdy^ ' dy^ Jx=x-^eh
the last term of which we will denote by B.

Now, if F{x,y) is a maximum, the first member of (1) is
negative ; and therefore its second member must be negative
also, and this whether h and k be both positive or both negar
tive, or either one be positive and the other negative ; and
whatever be the values of /i and k, provided only that they be
very small. If F{Xj y) is a minimum, the second member of

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MAXIMA AND MINIMA. 193

(1) must be positive under the same conditions and limitations
in respect to the signs and values of A and k.

But, when h and Jc are taken suflBciently small, the sign
of the second member of (1) will be the same as that of

d'F * dF

-^ h -f- 7- h, which must therefore be permanent and nega-
tive if F(x,y) is a maximum, and permanent and positive if
F{Xj y) is a minimum. It is plain, however, that the sign of

dF dF

- Jâ€” h-]- -f-k will change by changing the signs of h and k.

ax (ty

To make the sign of the second member of (1) invariable,

whether positive or negative, we must have

dF-, dF.

and, since h and k are entirely independent of each other, this

requires that

dF ^ ^dF ^ ,^,
^=0,and^==0 (2).

Let aj = a, y = 6, be values of x and y derived from these

equations, and denote by A, B, (7, Bu what ^^, ^^ â– ^, B,

respectively become when these values of x and y are sub-
stituted in (1) ; then (1) becomes

F{a + h,h+k)â€”F{a,h)=:~{Ah'' + 2Bhk-\'(Jk^) + B^ (3).

When the values of Ji and k are very small, and only such
values are admissible, the sign of the second member of (3)
will be the same as that of

Ah'' + 2Bhk + Ck\
which may be put under the form

-1+^11+3^

26

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194 DIFFERENTIAL CALCULUS,

The sign of this will be invariable, and the same as that of -4,
if the roots of the equation

A2 Bh C

are imaginary; being treated as the unknown quantity.
Solving this equation, we find

h^ A '

from which we conclude that the conditions for imaginary
roots are, that A and G have the same sign, and that the prod-
uct AC he greater than B'^.

In recapitulation, we say, that \{x = a,yz=h, make F(x, y)
a maximum then for these values of x and y we must have

dF_ ^_Q
dx" ^ dy '

d'^F d^F , , .

'd^' ^27 l>o^l^ negative,

dx^ dy'^ \dxdy) '
If x = afy =zbj make F{Xy y) a minimum, the conditions are

d^F d'^F
the same, except that then -j-^i ';r~2f '^^^st both be positive.

The existence of real roots for Eq. 4 indicates that we may
give such signs and values to h and k as to cause the expres-
sion Ah^ + 2Bhk -f- Ck^ to vanish, and, in so doing, change its
sign, whicli is incompatible with the existence of a maximum
or a minimum state of F{Xj y).

123. There remains to be examined the case in which
AOâ€” B^ =zO. When this condition presents itself, there may
also be a maximum or minimum value of the function.

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MAXIMA AND MINIMA. 195

By the theory of the composition of equations, or by in-
spection, the expression Ah^ + 2BhJc + Ck"^ may be written

{{4-'^1

^-1\(4 + b)'+ac-b-^

k^ / h \
hence, when AC â€” B^ = 0, this becomes -7 (-^7 +-S)> the

/ h V ,
sign of which, except when IA - {'Bj vanishes, is the same

as that of A; and i^(a, 6) is a maximum if ^ is a negative,
and a minimum i{ A is positive. Should lAj - \-B) vanish,

as it does when - = â€” -77 we cannot tell, without further
k A

inquiry, that F{a'\-h, b-\-k) â€” F(a, b) does not undergo a
change of sign. To decide this, let L, M, iV", P, represent the

d^F d'F d'F d'F . , ,

valuesof â€” , ^^^, ^^â€ž ^, respectively, when 0.^0,

y =zb; and also put Pj ^^^ ^^^ value of

\dx' ^ dx'dy ^ dxdy' ^ dy' A=^ + a/i

for the same values of x and y.

Introducing these values, and the conditions

^^z= 0, ^- = 0, Ah'' + 2i?M + (7i = 0,
dx dy ' '

the latter being a consequence of the hypotheses =^ = â€” j
and ACâ€” B^ = 0, we have
F(a -I- A, 6 + i) - J'Ca, 6) =

j-^ (iAÂ» + BMh^k + 3JVMÂ« -I- PiÂ») + Bj

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196 DIFFERENTIAL CALCULUS.

From this we see, that since the sign of F(a -|- A, 6 + i) ^
jP(a, 6), when h and h are very small, is the same as that of

jP(a, 6) cannot be a maximum or minimum, unless, if -^ i -{- B

vanishes,

Lh' + ^Mh^k + &Nhk^ + Pk'

vanishes simultaneously. Suppose these conditions to be sat-
isfied, then the sign of F{a -{-hjb-^k) â€” jP(a, b) is the same

as that of i?2- ^^t we have shown, that when A j^-\- B was

not equal to zero, and h and k were taken sufficiently small,
the sign of F{a + A, ft -f- k) â€” F{aj h) is the same as that of A;
but, when i^(a, b) is a maximum or minimum, the sign of
F{a + A, 6 4- *) â€” -^(Â«> ft) niust be invariable for all values
of h and k which are small enough to cause F{Xf y) to change
in value by continuous degrees in the immediate vicinity of
the value F{a, b). Hence it follows, that, when the values

h Ti

of h and k are such as to make - = -, these values must

k A

give ^2 ^ sign the same as that of A. If these several condi-
tions are satisfied, the function is a maximum when A is nega-
tive, and a minimum when A is positive.
124. If ^ = 0, J? = 0, (7 = 0, then

F{a + A, ft + fc) - F{a, b) =

-^ {Lh' + SMh'k + 3Nhk^ + Pk') + iZ^;

Ly Mj N, Pj i?2; denoting the same values as in the preceding
article. Hence, in order that jP(a, 6) may be a maximum or
minimum, L, Jff, N, Pj must separately vanish, and the sign of
^2 must be invariable ; and generally, when F{a, b) is a maxi-
mum or minimum, all the partial derivatives up to those of an

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MAXIMA AKD MINIMA. 197

odd order inclusive must vanish ; while those of the following
even order are so related, that the sign of the expression in
which they are the co-efficients of the powers and products of
h and k remains the same, whatever be the signs of h and k.
The conditions which will insure this invariability of sign
when the derivatives are of a higher order than the second,
are, in the general case, too complicated to be here discussed,
or even to be of much practical value.

12S^ Let it now be required to find the maxima and mini*
ma values of JP{Xj y, 2), a function of the three independent
variables x, ?/, z.

Referring to Eq. 7, Art. 121, and in the second member
stopping with the terms involving the partial derivatives of
the third order, denoting the aggregate of the remaining
terms by 72, and carrying F{Xj y, z) to the first member, we

have

i:t/ , X ,7 , -N rr/ . dFj . dFj . dF .

F{x + h,y + k,z-^i) - F{x,y,z)z=z~^-h + ~^^^^k-^ -^-i

Wlien A, &, i, have the very small values which alone are
admissible in our investigation, the sign of the second mem-
ber of (1) will depend upon that of the expression
dF, . dF, , dF .
dx'' + dy^ + 'dz'^'
and the sign of F(x -f- A, y + k, z + i) â€” F{Xj y, z) cannot
remain invariable for all the possible values and combinations
of the signs of /i, Zj, i, unless

dF. ^dF..dF, ^
dx^'+dy^+dz'^^^'^

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198 DIFFERENTIAL CALCULUS.

which IS, therefore, tlie first condition necessary to insure a
maximum or minimum of F(x,7/j z). But, because A, k, t, are
entirely independent of each other, the above condition in-
volves the three following : â€”

rf^_o ^^^-0 ^^-0 n

which will determine one or more systems of values for x, y, Â«.

And wo now proceed to inquire what further conditions must

be satisfied when one of these s^'stems, say x=za, y=:by z=zC)

renders the function a maximum or minimum.

Substituting these values in (1), and representing the val-

, . , d'F d'F d'-F d'-F d'F d'^F ^ ,,
ues which -^, ^^, ^^^^, ^-^, ^^, ^-^, i?, then assume,

by A, B, C, A', B', C, 7?,, respectively, we have

F{a + h,h-\- Jc, c + i)- F{a, h, c) =

â€” {Ah'' + Bh' + Ci"- + 2A'lik + 2B'lii + 2 Chi) + R^ (3),

the sign of the second member of which, when A, i, i, have
very small values, is the same as that of the expression

Ah'^ + Bh' + Ci' + 2Anik + 2Bnii + IC'hi (a) ;

and this sign must be permanent during all the changes in
the signs and values of 7i, fc, t, if F{a, ft, c) is a maximum or
minimum of F{Xj y, z).

Expression (a) may bo written

Make s =^ - ,t = -.\ then we have

i\A8'' + Bt^+0+ 2A'st + 2B\s + 20' t)]

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MAXIMA AND MINIMA. lOD

and the sign of this last will be the same as that of

or

/ B C A' B^ C^ \

The conditions that will make the sign of (b) invariable for
all possible values of s and t will make that of (a) invariable
for all the combinations of signs, and all admissible values of
hj kj i. To find these conditions, put (6) equal to zero, and
solve the resulting equation, with respect to either s or t,
say s. We thus get

Now, if the quantities A, B, C, A\ B', (7', have such relative
values, that the quantity under the radical, in this value of 5,
cannot become positive for any real value of t, then the paren-
thetical factor of (&) will always be positive, and the sign of
(6) will be the same as that of A, Putting this quantity un-
der the form

we see, that, to make it negative for all values of ^, it is neces-
sary and sufficient to have

A''' -AB<Q, i.e., AB -A'^>0 (c).

{A'B' - ACO'' < {B'^ -AC) {A^^ - AB) (d).

When conditions (c) and (d) are satisfied by the values of
Xj y, Zj deduced from Eqs. 2, F{aj 6, c) is a maximum or a mini-
mum ; for then the sign of expression (6), and consequently
that of the second member of Eq. 3, is permanent, and the
same as that o{ A. F(a, h, c) is therefore a maximum, if A is
negative ; and a minimum, if -^ is positive.

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200 DIFFERENTIAL CALCULUS.

Hence the conditions necessary for the existence of a mazi-
mum or minimum of F{x, y, z) are, that the values of x, y, Â«,
derived from the equations

dF_ dF_ dF_

ebould make

and

\dxdy dxdz dx^ dydz)

( /rf^Y_ d^ d^l (/^Y_ d'F^ d^l , ,,x
l\dxdz) dx' dz' \ \\dxdy) "" dx^ dy^\ ^ ^'

A necessary consequence of conditions (c^) and (d^) is, that

/d'FV d'F d'F d'Fd'F_/d'FV

\dxdz) dx' dz' ^ '^^ dx^ dz' ^\dxdz) -^

and hence -^, -^â€” ^, -i-j, must all have the same sign, which

18 negative when F{x, y, z) is a maximum, and positive when
F(x, y, 2) is a minimum.

126. If we have a function F{x, y, Â» . . . ) of n independent
variables, the first condition for the existence of a maximum
or minimum would be

dF^ . dF^ , dF . ^

^^+^^ + T.* + - = ' =

whence, because h,k,i...j are independent of each other,
dF dF dF

Eqs. 1 determine values z = a, y =zb, z = c . . ., which may
or may not produce a maximum or minimum state of the func-

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MAXIMA AND MINIMA. l-Gl

tion. To decide this question, we should have to examine the
term

1.2\dx'^ ^ dy' ^ dz' ^ ^ dxdy ^ )
in the expression for

F{x-^r1hy + h,z + %...) -^ F{x, y,z...).
If, for these values ot x, y, z . . ., the sign of this term is per-
manent, and negative for all admissible values, and all the com-
binations of the signs of //, i, i . . . , the function is a maximum :
if the sign is permanent and positive, the function is a mini-
mum. It would be found, that, to insure either of these states

d^F d'F d'^F
of the function, -r-^r? -7-7? -t-t""' must all have the same
dx^ dy^ dz^

sign, negative for a maximum, positive for a minimum. But
the investigation of all the conditions to be satisfied in this
general case, in order that the function may be a maximum or
minimum, is too complicated to fin^d a place in an elementary-
work.

127. Maxima and minima of a function of several variables
some of which are dependent on the others.

Let it be required to find the maxima and minima values of
the function u = F{Xj y, z. . .) of the m variables Xj y, z...^
which are connected by the n equations

f^{x,y,z...) =

J

(!)â€¢

By eliminating from w, n of the m variables, by means of the n
given equations, u would become an explicit function of m â€” n
independent variables, and its maximum or minimum could

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202 DIFFERENTIAL CALCULUS.

then be found by the method just explained ; but this elimi-
nation may be avoided, and the determination of the maxima
and minima of u greatly simplified, by the process which fol-
lows : â€”

Suppose the variables x,y,z.,.,io receive the respective
increments A, i, f . . ., by virtue of which the function passes
from a given state of value to another in the immediate vicinity
of this : then, if the given state be either a maximum or a
minimum, we must have

dF, dF^ dF . ^ ,^,

The partial derivatives of the first members of each of
Eqs. 1, taken with respect to each of the variables, are separate-
ly equal to zero. Taking these equations in succession, multi-
plying- each partial derivative by the increment of that varia-
ble to which the derivative relates, and placing the sum of the
results equal to zero, we have

l' + f* + f'> -n (S,

There being a number (n) of Eqs. 3, these with (2) make n -f- 1
equations of the first degree in respect to the m quantities
hjkj i . , .: hence, by the combination of these equations, wo
may eliminate n of these quantities, and arrive at a final equa-
tion involving the remaining m â€” n quantities, and also of the
first degree with respect to them. To facilitate this elimina-
tion, let fij, f*2Â»-'7f*n; denote undetermined quantities; and

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MAXIMA AND MINIMA, 203

multiply the first of Eqs. 3 by f^i, the second by fÂ«2 â€¢ â€¢ â€¢; ^te
n*"^ by fiâ€ž ; add the results to (2), and arrange with reference
to hj k,i ...: we thus get

+ Vrfy ^''^ rfy ^''^ dy + +''â€¢ ^

= (4);

+

a true equation when F{Xj y^z ,, ,) is susceptible of a maxi-
mum or minimum, whatever be the values of ^ui, ^2 â€¢ â€¢ â€¢> I'^n*

Place the co-efiicients of n of the quantities /i, h, i . . ., in
Eq. 4, equal to zero : the n equations thus obtained Avill deter-
mine f*uf*2Â«Â«M ^w Ey substituting these values of fti,iW2...,/Â«â€ž,
in (4), n of the quantities hjk,i. ,,, will vanish from that equa-
tion ; and, if the co-eflScients of the m â€” n of these quantities
remaining in the equation be placed equal to zero, we have,
including the n given equations,

/,{x,y,z.,,) = 0.,./â€ž{x,y,z..,) = 0,

m equations from which to determine values a,h^ c, ., for the
m quantities Xj y^z,,,j respectively. This is equivalent to
equating the co-eflScient of each of the m quantities /i, k,i . , .,
in (4), to zero ; and these m equations, together with the n
given equations, will make m-\-n equations, by means of
which we may eliminate the n indeterminates f^ij 1^2** '7 hm
and find the m quantities x^y^ z ...

It remains to be ascertained whether the sign of the expres-
sion for F{a -\- h, b -\- kj c -{- i . , .) â€” F(aj 6, c . . . ) is invariable ;
and, if so, whether it be positive, which answers to a minimum;

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204 DIFFERENTIAL CALCULUS.

or negative, which answers to a maximum. Theoretically,
this examination is very complicated ; but, for most cases in
wliich this method is applicable, the form of the function en-
ables us to decide at once Avhich of the two states, if either,

When m â€” n = 1, or there is only one more variable than
there are equations connecting them, the case discussed in this
article reduces to that of an expression which is implicitly a
function of a single variable.

128. In the case in which it is required to determine the
maxima or minima of a function, the several variables of which
are connected by but one equation, the process may be still
further simplified.

Let u =. F{x, y, 2 . . . ) be the function, and
/ix,y,Â»...) = (1),
the equation expressing the relation between the variables
x,y,z... : then, by the reasoning employed in the last article,
we have

|*+|. + |.+ ...=0 (3,

Multiplying (3) by tho indeterminate /*,, subtracting the result
from (2), and arranging with reference io h,k,i..., we have
/dF d/\, , /dF d/\. , IdF d/\. , â€ž ,,,

[d^-''Â£r+{d^-%}+{d.-dy +â– â– â– '=' w-

Equating to zero the co-efficients of the several quantities
A, Aj, i . . . , we should have, with the given equation, m + 1 equa-
tions, by means of which we can eliminate fi, and determine the
m quantities. But from the co-efficients of A, ft, t ... , in (4),
placed equal to zero, we find

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EXAMPLES OF EXPANSION.

205

f* =

dF dF dF
dx dy. dz

that IS, the ratio of the oo-eflScients of A, in Eqs. 2 and 3, is
the same as that of the co-efficients of Aj, of i . . . This rela-
tion will be found to facilitate the determination of maxima
and minima.

The examples which follow are arranged in the order of the
articles in this section under which thev fall.

{x+hYia + y+ky=

EXAMPLES.

1. I(F(x, y) z=^ x'^{a-\- yf, find the expansion of

{x + hy{a + y + ky
in the ascending powers of A and k.

'^\a + yy+2x{a + yyh+3x\a+yyb
+{a+yyh+6x{a-{-!/y/ik+Sx\a+y)k^
+ S{a + yyh^k + 6x{a + y)hk^ + x^k^
+ S{a + y)h^k^ + 2xhk' + h^kK

2. If F(x, y, z) = ax' + by' + cz' + 2exy + 2gxz + 2/yz,
find the expansion of F(x + A, y + &, Â« + i).

Â«a3^ + 6y ^ + C2 ^ 4" 26xy + 2gxz + 2fyz
+ 2{ax + gz + ey)h + 2{by +/z + ex)k
+ 2{cz+/y + gx)i-{-{ah^ + bk^+ci')
+ 2{fki + ghi + â‚¬hk).

3. Expand

in the ascending powers of x and y.

F{x + h,y + k,z + i):=^

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206 DIFFERENTIAL CALCULUS.

+

4, Find what values, if any, of x and y, will render the func-
tion F{x, y) =. x^y -{â–  xy'^ â€” axy a maximum or minimum.

From the equations -^ = 0, tâ€” ==â–  0, we get four systems
of values, viz.

x = 0^ x = a^ x = 0) ^ = 3
y = 0)' y^of y = a)' y=z|['
none of the first three of which satisfy the condition

d^^dr/{d^yj ^^''â€¢'''^'

and must therefore be rejected. The fourth system reduces
this inequality to ^a^ ^ â€” > which is true, and at the same

time makes both ,-^- and -, 3" positive : hence the values
a; = ~, y =Qj Daake the function a minimum, and this mini-

o o

a'
mum IS â€” ^Â»

5. Determine the values of x and y that will make

F{x, y) = e-^'*+Â«''>(axÂ« + bif)
a maximum or minimum.

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EXAMPLES,â€” MAXIMA AND MINIMA. 207

3â€” z= 0, -^â€” = 0, give the three systems of vahies
ax dy

y = ^\' y^^\) y = y
which we will examine in succession.

The first system gives _ = 2a, ^ = 26, ^-^ = 0:

hence the values ar = 0, y = 0, make the function a minimum.
With the second system, we have

hence the existence of a maximum or minimum depends on the
relative values of a and 6. If 6 is greater than a, -jâ€”^^^i ^-^^
have the same sign, which is negative, and the function is a max-

Jl jp J2 Jp

imum; but, if ft be less than a, ^ .^, -j-r, , have opposite signs,

and the second system of values of x and y make the function
neither a maximum nor a minimum.
For the third system, we find

from Avhich we conclude that a? = it 1, y = 0, will make the
function a maximum when a^h; but, when a<6, it has
neither a maximum nor a minimum.

6. The equations of two planes, referred to rectangular co-
ordinate axes, are

/,{x,y,z)=Ax + By+Cz-^D=zO (1),
/,{x,y,z)=zA^x + B'y+C^z-D'=:0 (2).

It is required to find the shortest distance from the origin of
co-ordinates tct the line of intersection of the planes.

Let F{x,y,z) = x'^ + tf + z^ (3)

represent the square of the distance from the origin to the

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