Now, since F(Xjy), F(x*jy')^ differ only by having a: and y
in the one replaced respectively by x^ and y' in the other, it
follows that any partial differential co-efficient of i^(a;, y) will
be the same function of x and y that the corresponding partial
differential co-efficient of F{x\ y') is of x' and y' ; and hence
the hypothesis that renders a; = a;', j^ = y ', will, at the same
time, cause these differential co-efficients to be equal. There-
fore make a? = aj' = a + //<, y = y' = 6 -f- fc^, and we may
write
/c«)(0 =
rfa;" dx^'-Hy
/i»-ii + .,
+ 71—- AA;»""^+- — i"
dxdy^"^ dy^
Wj
Digitized by VjOOQIC
INDETERMINATE FORMS. 167
all of the several orders of partial differential co-efficients of
F{x, y), up to and exclusive of the n***, vanishing for x = a^
y = bj that is, for ^ = in f{t) . . ./"-^^ {t) ; but all of those
of the n^ order not vanishing. Then, writing Gt for t in
Eq. 4, and substituting in Eq. 3, we have
F{a J^kt,h + U)- F{a, b)
n\dx'^ ' dx'^-^dy
d'^F _ d^'F \
and if, in this equation, we make ^ = 1 ; then
F{a + A, 6 + i) — F{a, h)
which enunciates a theorem relating to a function of two inde-
pendent variables analogous to that demonstrated in Art. 56
for a function of a single variable.
In Eq. 5, suppose both a and h to be zero, and then change
h and h into y and x, as we may do, since h and h are not only
independent of each other, but may have any values, and we
have
which expresses another theorem relating to a function of two
independent variables similar to that in Art. 66 for a function
of a single variable.
Digitized by VjOOQIC
168 DIFFERENTIAL CALCULUS,
111. Let F{x, y)j /(x, y), be two functions of the inde-
pendent variables x and y, and suppose that not only the func-
tions, but also all of their successive partial differential co-effi-
cients, up to those of the {n—iy^ order inclusively, vanish for
X = aj y =^ b ; but that, in respect to those of the w*** order, all
do not vanish, nor do any of them become infinite for these
values for x and y: then by Eq. 5, Art. 110, remembering
that by hypothesis JP(a, b) = 0, /(a, b) = 0, we have
F{a + h,b + k)
1.2.3. ..n\dx" dx"-^dy '
d"^ 77« , . ^"^7„\
Dividing (1) by (2), member by member, we have
/(a + h,b + k)
(d^F d*F d'^F d*P
(3).
Now, the increments h and k are quite arbitrary, and, like
the variables to which they refer, are also independent of each
other : we may therefore assume k = mh, in which m is an ar-
bitrary constant.
Substituting this value for ft, in Eq. 3, dividing out the
factor A", common to the numerator and denominator of the
second member, and then making A = 0, we have
Digitized by VjOOQIC
INDETERMINATE FORMS.
169
F{aJ>) _
/(a,6)-0
d-F , d'F , , d"F
d'F
d'f , d'f , , d'f . , , d'f .
(4).
This value of J) ^ , ^ = - is indeterminate, since m is arbi-
f{a,b)^
trary; and generally, if two functions of two independent
variables both reduce to zero for particular values of the
variables, the ratio of the functions for such values is really
indeterminate.
112. Making n = 1, in Eq. 4 of the last article, we have
F{aji)
f{a,b)
dF dF
dx ' dy
dx'^ dy
and this value of }-, * A becomes determinate if -j-, ~, both
/(a, 6) dx^ dx
vanish for a; = a, y = 6; or, they remaining finite, if -,-, ^-, both
vanish for these values of x and y. The value of - ^ , \[ = -
/(a,i)
becomes, in the first case,
dF . dF^
-^)-' - ; = -~; and, in the second, ^; \ c = -^.
dF dF[
,; \( is also determinate if -,-^ = -^ : we should then have
/(a, 6) df dj^
dx dy
dF/df^ df^ \ dF dF
JP(a,6) _ dx \dx "^ rfy / _dx __dy
A<^.^'' f_(df dy^\^ df- df
- . "T ^,, / da? dy
22
Digitized by VjOOQIC
170 Differential calculus.
I^di- = ^' Jx = ^' ^ = 0, ^^^ = 0,wemaken = 2inEq.4,
Art 111, and thus have
d}F <r-F d}F ,
dx^ dxdy dy'^
which is indeterminate, except in particular cases depending
on the absolute and relative values assumed by the partial
differential co-efficients for the values a; = a, y = 6.
Example, z =. — ,— ^ ^ ^ = - when x = 1, y = 1.
^ a; + 2y - 30
Here JP(a;, y ) = ite + Zy, f{x, y ) = a? + 2y — 3,
^^-^-1 forx-1. ^^-1
;-—=-= 1, for y = 1 ; J- = 2; hence
' >
_ l+m.
and therefore, for the assigned values of x and y, the function
is really indeterminate, and may take any value between + oo
and — GO .
113, In the case of the implicit function u = F{x, y) = 0,
we have found
dF
dy _^ dx
dx^^dV (1)-
dy
Now, if aj = a, y = 6, are values of x and y, which, while they
dF
satisfy the given equation, at the same time make -r- =0,
Digitized by VjOOQIC
INDETERMINATE FORMS. 171
-V— = 0, then -^ takes the indeterminate form - , and its true
dy ax
value, if determinate, must be found by the preceding method.
DiflFerentiating numerator and denominator, we have for
x=ayy=zbj
dF dP-F d'F dy
^ dx dx'^ dxdy dx ^2)
dx 'dF^^WF d'F~dy ^ ^'
dy dxdy dy^ dx
from which we get
dy'' \dx) ^ dxdy dx^ dx'' "^ ^'
a quadratic with respect to -,- This equation agrees with
dF
Eq. 3, Art. 96, observing that by supposition ^ - = 0. It must
be remembered that Eqs. 2 and 3 are true only for the partic-
ular values X z=:a,y =ib. When these values of x and y, in
addition to making the function and its first partial diflferential
co-efficients equal to 0, also make
^_0 -^-0 ^-0
dx* ~ ' dxdy "" ' rfy* ~ '
the value of -^ , as given by Eq. 2, again takes the form - ;
uX \j
and we must in that case effect a third differentiation, which
gives
dy __ dx^ dx^d y dx "^ dxd y^ \dx ) "^ Ix dy dx^ .,.
~di-^ d'F _ d'F dy d^F7d^f~d^F'd^ ^^^'
dx'dy dxdy^ dx dy^ \dx) dy'' dx''
d'F fl'^W
and from this, observing that by hypothesis -^—-i- = 0, -r-j = 0,
we derive the cubic equation,
Digitized by VjOOQIC
172 DIFFERENTIAL CALCULUS.
^ W+ 3 i!^ m+ 3 -1'^ ^ + ^= (5).
dy^ \dxj dxdy^\dx/ dx^dy dx dx^
dy
Ex. 1. Determine the value of -^ from ax^ —y^ — by^=z
dx
when x=zOj y = 0.
Here -/ = ^ , ^^^,. = - for a; = 0, y = 0.
rfx 3y^ + 2by
But, for these values of a; and y, we have
rfy 2ax 2a 2a « ^ ^
da; da; dx
. <^_l__±_ f^\—a dy _ la
dx
Ex. 2. If w = a;* + 3a«a;« - 4a''a;y — a«y» = 0, find the
value of -^ for a; = 0, y = 0. We have
dx
^" = 4x» + Ga'a; - 4oV.
• CLX
-^=- z= — 4a^a? — 2a^y:
dy ^
dy __ 4x^ + 6a^a; -> 4a ^ _ 2x^ + 3^20; _ ^g^y
dr"" • 4c?a; + 2a2y 2a'x + a^y
= - for a; = 0, y = 0.
Differentiating both numerator and denominator with respect
to X and y, we get
6x' + Sa'-2a'i^ Sa^ - 2a^ ^
dy ^ dx dx ^ ^ ^
-T- = :^z = :t- for a; = 0, y = 0,
dx dx
3-2 f-
dx
^+1'
Digitized by VjOOQIC
INDETERMINATE FORMS. 173
2^ + ('^\=S-2^^
(I)' -
dx \dx) dx '
ix:)+*|-3=o,
dy
Should it happen that the particular values of x and y reduce
d'F ^., d'F d'F . ^ . . ^
— — - to zero, while — — — j — — - , remain finite lor these values,
dy^ d±dy dx-
dy
then Eq. 3 of this article gives, for one of the values of — ;
CLX
dr-F
dy _ dx'^
■dx~~. d'F '
2i
dxdy
which is finite, while the other value of ,^ becomes infinite,
dx
as may be shown by discussing the equation ax'^-\-hx -\- c=0,
under the suppositions that a = 0, and that h and c are finite.
114i. The investigation of the true value of ~-^. when it
dx
takes the form - for a; = 0, y = 0, may be simplified by the
consideration, that, in this case, (-7^ L_o = ( - L^o; as is evi-
dent from the definition of difierential co-efficients. Take Ex.
2 of the preceding article, and divide through by x'*" ; then
X \x/
y
Solving this equation with reference to - , and then making
a: = 0, we find, as before,
dx x
Digitized by VjOOQIC
174 DIFFERENTIAL CALCULUS.
In like manner, the example
x^ + ^y^ ~~ 2axy^ — Zax'^y =. 0,
which gives
dy__ 4.x' - 6axy ^ 2ay^ _ x- q q
dx - Sax^'^^+lcucy - Say' - q ^ "^ ^ "' ^ " "'
by dividing through by x'y takes the form
\X/ \XJ X,
a cubic equation, from which, after making a? = 0, we get for
- the three values 0, 3, and — 1.
X
For another example, take the equation
x^ + ^^^y + ^^y^ — y* = ;
y (y\} [y\}
whence x-\- a - \-h\-\ — y(-) = 0,
X \xj \x/
= 0j y =:0j reduces to
X \xj
11 '11 a
5 we have - = 0, and - = — -•
x X b
through by y', the assumed equation becomes
^©"+"©'+'^''="'
;isfied by making simultaneously a? = 0, y = 0,
^ y
3 " = 0, or - = oo , will satisfy the given equation
y X
I with the values x = 0, y = 0. Therefore, when
y
e these values, - may have the three values,
X
Digitized by VjOOQIC
INDETERMINATE FORMS. 175
EXAMPLES.
1. ( ^ — ^ ) Ans. -•
2- f ' — r"' ] Ans, — -
sin.»a; /^^
3. / e - 2sin..-c-A ^^^ ^
\ x— sin.a; /x=o
*• t"! ) Ans. 0.
In solving this example, begin by making re' = - ; whence
z
1
2 = oo when a; = 0; and we conclude that ^"'* decreases
.more rapidly as x decreases than does a?^**, however great be
the value of n.
5. ( . =r Ans. — 1.
/ nx t nx . nx . , nx\ —
6. / g, +» ,_ j-a, +...+ a\. Ans.a,a,a,...(v
\ « /,=o
7. / J-l+_(^-l)h ^„3_ 0^
\ 's/x^—i /x=i
8. f 2'8in. — j Ans. o.
V — cos.nx/^^^
9. { , = ) Ana. ~
10. 1 + - Ans. 1.
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SECTION XI.
DETEBMINATION OP THE MAXIMA AND MINIMA VALUES OF FUNC-
TIONS OP ONE VARIABLE.
115 • When the valuo of a function, for particular values
of the variables, is greater than those given by values of the
variables immediately preceding or following such particular
values, the function is said to be a maximum : when it is less,
it is a minimum. To fix attention, suppose y =i/(^x) to be
such a function of ic, that, as x gradually changes from a spe-
cific value to another, y undergoes continuous changes ; but,
having increased up to a certain value, then begins to de-"
crease, or, having decreased to a certain value, then begins to
increase. The value of y at the point where, from. increasing,
it begins to decrease, is a maximum ; and at the point where,
from decreasing, it begins to increase, it is a minimum. In
the first case, the value of y is greater, and in the second case
less, than those which immediately precede and follow. The
terms maximum and minimum must be understood as relative
rather than absolute ; for it is plain that a function may have
several maxima and minima as above defined.
Confining ourselves, for the present, to explicit functions of
a single variable, we have seen (Art. 52) that such function
can pass from increasing to decreasing, or the reverse, only
when the first differential co-eflBcient of the function passes
through or 00 , or when this differential co-eflScient changes
from positive to negative, or from negative to positive.
176
Digitized by VjOOQIC
MAXIMA AND MINIMA, 3 77
Hence those values of x which render y =/{x) a maximum
or a minimum must be found among those which satisfy the
equations /'(re) = 0, /'(x) = oo .
Let a; = a be a root of one of these equations, and let A be a
very small quantity ; then /(a) Will be a maximum if/'(« — A)
is positive, and /'(a + A) is negative ; but /(a) will be a mini-
mum if /'(a — h) is negative, and/' (a + A) is positive. When
/'{a — A), /'(a + A), are both of the same sign, whether posi-
tive or negative,/(a) is neither a maximum nor a minimum,
Ex. 1. y =/{x) = lax — x'^,
f'{x) = a — x; f(x) ■=! gives a; = a,
f'{a — A) = a — a + A = + A>
f*(a + A) = a — a — A= — A.
Hence a; = a renders the expression lax — a;^ a maximum, as
may be easily verified ; for, making x=ia^ 2ax — x^ reduces
to a^ ; but making a; = a + A, or x = a — h, our result in
either case is a^ — A^ < a^.
116. The method just given for deciding whether or not
a root of the equations /^(x) = 0, /^(x) = oo , answers to a
maximum or minimum state of/(ar), is general ; but, in respect
to the roots of the equation /' (a;) = 0, we may for this pur-
pose deduce a rule, that, in many cases, admits of easier appli-
cation.
As before, let a; = a be a root oi f'{x) = 0, and suppose
that f^^\x) is the first among the derivatives oif{x) that does
not vanish for this value of x; then (Art. 56)
/(a + h) -/(a) = j-^/("'(a + eh) (1).
Since ^ is a proper fraction, and A, as we shall suppose it to
be, is a very small quantity, it is obvious that the sign of
/^"^(a-j-^A) cannot change with that of 7i, and is therefore
23
Digitized by VjOOQIC
178 DIFFERENTIAL CALCULUS.
invariable : hence the sign of the second member of Eq. 1 de-
pends on that of A" when combined with that of /<"^(a + Oh).
But, if n is an even numoer, the sign of /** is positive, what-
ever be the sign of A; and in this case the sign of the second
member of Eq. 1, and consequently that of /(a + h) —f{a)j
will be the same as that of /<'*^(d + ^A), or as that of/^"^(a),
since f^^\a + 6h) and/^"^(a) have the same sign. If, then, n
being even, /^"^(a) is positive, /(a + h) —/{a) is also positive,
whether h be positive or negative, which requires thaty(a) be
less than /{a zb h) ; that is, /(x)^^„ =/(«) is less than those
values oif{x) which are in the immediate vicinity of this par-
ticular value. This condition indicates a minimum state of
the function. But, n being still an even number, if /^"^(a) is
negative, then /(a rb A) — f{o) is negative, which requires
that /(a db h) be less than /(a) ; and a maximum state of the
function is indicated.
The hypothesis in respect to h being continued, if n be an
odd number, then, since ( + A)'* and ( — hy have opposite signs,
and the sign of /"(a + dh) does not change with that of A, the
second member of Eq. 1 will change its sign as h changes from
positive to negative, or the reverse. Hence f(a -f- h) — /(a)
and /(a — h) — f{a) must have opposite signs, and f{a) is
greater than one of the expressions /(a + /i), /(a — A), and
less than the other ; that is, /(a) is neither greater than both
the immediately preceding and immediately following values
of the function, nor less than both these values; and therefore,
in this case, x=: a renders the function neither a maximum
nor a minimum. Whence the rule for deciding which of the
roots o{ f^{x) = corresponds to maxima or to minima of /(x).
" Substitute the root under consideration, in the successive
derivatives of the function, until one is found that does not
Digitized by VjOOQIC
MAXIMA AND MINIMA, 179
vanish. If this derivative is of an oven order, and the result
of the substitution is positive, the root will render the function
a minimum ; but, if the result of the substitution is negative,
the root will correspond to a maximum* If the first of the
derivatives of the given function that does not vanish is of an
odd order, the root corresponds to neither a maximum nor to a
minimum state of the function.''
Ex. 1. Find the values of x that will render
/{x) = x^- dx^ -u 24a: — 7
a maximum or a minimum,
/' (x) = 3x^-^lHx+24:=z 3(a;2 — 6x + 8),
f^{x) = 6{x-S),
From f{x) = 3(a;2 _ 6x + 8) = 0, we find a; = 2, or a; = 4.
When x = 2, f"{x) = 6(x - 3) = — 6 ; and hence, for x = 2,
the function is a maximum. When x = 4, f"{p^) = + ^ ; and
a? =z= 4 renders the function a minimum-
Jiy, When y is an implicit function of x given by the
equation F(x^ y) =. 0, and the values of x corresponding to
the maxima or minima values of y are required, we may, in
cases in which the resolution of the equation with respect to
y is possible, employ the methods of the preceding articles.
But, without solving the given equation, we may proceed as
follows : —
Let u := F{Xj y) = ;
then (Art. 84)
du
dy dx
dx du
dy
Limiting our discussion to the values of x derived from the
equation -~ = 0, if -=- is finite, ~z=0 requires that -,- =0.
^ dx dy dx dx
Digitized by VjOOQIC
180 DIFFERENTIAL CALCULUS.
Hence we have the two equations
by the combination of which, eliminating y, we get a single
equation, in terms of x, which will determine those values of x
which may or may not render y a maximum or minimum.
To decide this, we must pass to ^-f^, which, since -^- = 0, re-
duces to
d'^u
d'^y ^ dx^,
dx' da '
dy
and if the values of a; and y derived from the equations w = 0,
du
-— =r 0, do not cause this to vanish, but make it positive, the
CiX
value of y corresponding to this value cf a? is a minimum ; but
d'^y
if, by these substitutions for x and y, -~ becomes negative,
the value of y is a maximum. But, if these values of a; and y
cause -j-^ to vanish, -^-^ must also vanish in order that y
may be a maximum or minimum ; and it would be necessary to
d^v
find ^^ , and substitute in it, to enable us to decide whether
y is a maximum or a minimum.
Ex. 1. Find the value of x that will render y a maximum or
minimum in the function
u = x^ — Zaxy -{-y^ = (1).
du
dx
dx ay — x^
du y' — ax
dy
Digitized by VjOOQIC
MAXIMA AND MINIMA. 181
-^ = answers to ay — iB* = (2).
From (1) and (2), we find
therefore rr = 0, or a; = a 4^2, and the corresponding values
of y are y = 0, y = aA^A.
The values a; = 0, y = 0, in the expression for -^- , cause it
to take the form - ; and the true value of -~ must be found.
ax
This may be done by the method of Art. 113. It is better,
however, to proceed as follows : —
The second and third derived equations of the given equa-
tion are
(^■-)g+(«^l-^)3'+<l)'+^=«-.
du
and when, in these, we make a; = 0, y = 0, we find -^ z= Oy
and -y-^ = — • Hence y = is a minimum when a; = and
y =: 0. When x = (is/% ^he corresponding value of y = a-^4
is a maximum ; for we have, in this case,
^ _ _ dx^ _ 2a? 2 a ^2 _ _ 2
rfa;2 "" -^ — -y'l^ax^ a'^\^ — a'^2~' a'
dy
which indicates a maximum.
118. Suppose that the relation between a?, y, and w, is ex-
pressed by the two equations u = F(Xy y), /(a;, y) = 0, so
that u is implicitly a function of x ; for, deducing the value of y
Digitized by VjOOQIC
hence
182 DIFFERENTIAL CALCULUS.
in terms of x from /(x, y) = 0, and substituting this value of
ymu = F{Xj y), we should have u an explicit function of x.
If the maxima and minima values of u were required, we
might pursue this course; but we may accomplish our purpose
without solving the equation/(ir, y) = 0.
We have (Art. 82)
du_dF dF dy
dx dx dy dx
also (Art. 84)
d£
^y^_ dx
dx (Jf •
dy
du __ dF dF dx
dx dx dy df
d'y
Now, the values of x and y which satisfy simultaneously the
equations f{x, y) = 0, and -,- = 0, or the equivalent of the
CtiX
latter,
dFd/ _dF^_^
dx dy dy dx'^ '
d'^u
but which do not cause -r-r to vanish, will render u a maxi-
dx^
d^u
mum or a minimum, according to the sign of -j-j' But, if
CbX
d^u
-j-^ vanishes for these values of x and y, we must, as before
explained, pass on to the derivatives of the higher orders, to
enable us to decide the question.
Ex. 1. Given
u^x' + y^ = F{x,y) (1),
(x^ay + {y^by^c^ = 0=./{x,y) (2),
Digitized by VjOOQIC
MAXI3IA AND MINIMA, 183
to find the values of x and y that will make u a maximum or
minimum. We find
and therefore
^df _dFd/^^
dx dy dy dx
becomes
x{y-'b) — y{x-d) = 0:
whence ay = bxy y=i-x.
Substituting this value of y in (2), we have
a;2 _ 2aa; + a2 + *!a;2 - 2 ~ a; + ft^ — c' = 0,
or
whence
a;2^1+^)~2x(a + *-') + a^ + 6^-c2 = 0:
ac
a; = a nb —7^^ .
\/a* + 6^
By differentiation, we get from Eqs. 1 and 2
d}u_ 2bc''
Observing that the upper sign in the value of x answers to
d'^u
the upper sign in the value of -,^, and the lower sign in the
one to the lower sign in the other, we discover that
x =z a A- — 7
d'^u
renders -^-^ negative ; hence this value of x makes u a maxi-
mum : but, when a , is substituted for x, -^-^ be-
Va^ + i^ dx^
Digitized by VjOOQIC
184
DIFFEREXTIAL CALCULUS,
comes positive, and u is therefore a minimum for this value
of X.
119. When we have n variables connected by n — 1
equations, by processes of elimination, we may reduce the
n — 1 equations to a single equation involving but two of the
variables, and thus bring the investigation of the maxima and
minima of the variable which is taken as dependent to the
case treated in Art. 117. But, in general, it will be found
easier to operate as follows : —
Suppose that the four variables, x, y, z, u, are connected by
the three equations,
/,{x, y, z, u) = 0, /2(x, y, z, u) = 0, /,{x, y, 2, w) = ;
and that the maximum or minimum value of u is required, x
being the independent variable. Differentiating with respect
to Xj we have
dx
dx
dz dx
,^fidy df2 dz ^^, r r :^
dy dx dz dx ' du dx
da dx
df^ du
dfz dy df^dz df^ du
L Z<J IjL I
dx dy dx^
=
(1).
dz dx ~^ du dx
One of the conditions for a maximum or minimum for u being
du
^- = 0, introducing this in Eqs. 1, they become
dA
dx
df,
dy dx
d/^dy
dz dx
d/2 dz
1 , ./i C/ J
dx dy dx^^ dz dx
=
df^ dy
dy dx
dx ^^' f^^^^ ' '
df^dz^
dz dx'
(2).
The equation which results from the elimination of
dy dz.
Digitized by VjOOQIC
MAXIMA AND MINIMA. Ic5
from Eqs. 2, togetlier with the three given equations, which we
will denote by fi = 0, /2 = 0, /j = 0, will determine values of
Xf y, z, and u. To decide whether any or all of these values,
or rather systems of values, make u a maximum or minimum,
d'^u
we must ordinarily pass to -r— 2, and find what sign it takes
when the values of the variables are put in it. By difFeren-
d^u
tiating Eqs. 1, the resulting equations and Eqs. 1 will give -j—^.
120. Before concluding the subject of the maxima and
minima of implicit functions, we will briefly refer to the limi-
tations made at the beginning of Art. 117. Resuming the
equation
du
dy
dx^
dx
'du'
dy
we
remark.
that
the
necessary
condition
for
a
maximum
or
minimum value of y is, that — change its sign, which it can
do only when it passes through the values or 00 . Now,
dy 1 ^^ /x ^^ 1 . /. .
-^ becomes zero when — z= 0, - bemff nnite ; or when
dx dx dy ^
— ziz 00 , -— being finite. Again : ^- becomes infinite when
dy dx dx
du ^ du . . £ .. , du ^^ u ' XI '^
-— = 0, -7- remainmg finite : or when — - zz: 00 , - bemg finite.
dy dx dx dy
It therefore appears that the methods heretofore given for
determining the maxima and minima of implicit functions are
quite incomplete, as they omit the discussion of several cases
that may give rise to these states of value.
Most of the functions with which we have to deal are those
24
Digitized by VjOOQIC
186 DIFFERENTIAL CALCULUS.
whose maxima and minima are indicated by a change in the
sign of the first derivative when it passes through zero. It
often happens that the conditions of the problem to be inves-
tigated enable us to decide some of the questions relating to
maxima and minima, which, if referred to general rules, would
require great labor.
EXAMPLES.
{When rr = - , t^ is a
minimum.
1.
1 - a; + a;2
2.
X
\ -\- X^
3.
t^ = ^' + 2cos.a: + c
4.
1
u^x'-^.
{ When a? = 1, w is a max.
( ^' a; i== — 1, w is a min.
When a; == 0, w == 4, a min.
A max. when x^=e.
5. Divide the number a into two parts, such that the pro-
duct of the m^^ power of one part and the n^^ power of the
other part shall be a maximum.
Ans. <
The parts are , — and — ; — ; and
m-\-n m-\'n^
their product, m" n" ( ) when
\m-\-nJ
m and n are even numbers. The prod-
uct may also have two mimimum
^ states.
6. Find a number such, that, when divided by its Napierian
logarithm, the quotient shall be a minimum.
X
The function to be operated with is .
LX
Ans. a; = c.
Digitized by VjOOQIC
MAXIMA AND MINIMA. 187
7t
I. u = sin. x{l + COS. x). ' A max. when a? = -•
o
X
8. tt = :r—, 1 A max. when x = cos. a?.
1 -[- a; tan. ic
9. Find the number of equal parts into which a given
number a must be divided, that the continued product of these
parts may be a maximum.
{Each part must be e, the number of
parts -, and the product {ey .
10. Of all the triangles standing on a given base, and hav-
ing equal perimeters, which has the greatest area ?
Denote the base by 6, and the perimeter by 2p, and one of
the two unknown sides by x.
^ 2p — & (The triangle
.^ns. X — -^ — • K , , _
^ (IS isosceles.