-
9I
x
,ZI
9- oZ
toz- -
D +9- lr* !-l urur 'D-: x 'e'r0: I+rrueqlv[ (*rT I II Ir E
i-1.i-1" s
'u 3 x rcg9 =
@) ! oroJerer{lE
'x >
x>I'I- !
'xE
'x-Z'xE-
_ lr + xZ- l+ lr +xl _ @){ #Z-: o:uauu-udq(I)'gf
- |unr'lt*,'.' 'S
-J YZ 1-c
-CCC)rv ltz uEcS
.0I-: - uul .-vl vts urr+ I u,l'sunr+lunt)iz )-cv v ut
.9.- 0ZZl\/,tt/\/rro --- o e 9-: 0-)t'lo-Il
- 0)tuaqt'I-_ruerlv\ (l)IE
'O # o etoleroql 'I sl "[ lo enlelruruurunueqro'r'T _ 0-){ puu
I
= I + lt + xl
-
(x){ ueqt'0 _ p ueq/V\'0
- I + xD to 0 - I + f''e'l '1urod Eururnl oruos ]u sl qdur8 eql
ueq/y\ ue>le I s\Z anlel ruruuruFu eql
EoroJeroql 'eug 1euo3,(1od u s\ { go qder8 eqt 'o tequtnu IBer
uea.rE ,(uu roC
(y-g) uollsenO EupseII suollsan} Eupsa; ol suollnlos
qq'uerrord sl (U) '0I
-
L ,uat + a ,LrEl'.')tzc yo
u)tzuvzSS995
-Ir?J Jel UBl . -
Uel Uel . -
UUl(' y(' uZ y yZ u'(VE'g1) uorg
perdwf,lg pcqeuraqtery uO seroN arruce'I
's -
*"o :iuel' s--*
r t' S I ) Jo sloor rnoJ erll oru V ' t' Z' | :'0-9+zx1l-v:
seruosoq u uau
'o-(s+zxol-vx,x
- T zxE - I
'{1}uelentnbe to'c :xz- {-xt.0zwt-l: 0zwlt?uelz- 0 ewt -
uotlenbe eql soUSIIBS ]l 6eBInuIJ(; '!T,rrel_ : Ttwt uotlenbe
e{} segsl1I
rt&s
-
-
Il.* I.Utr
rffi
'x>z 'xf; )'z),.r::. '.ii:i : l'* .1_.l +rr+rl _ (x)t
Z--DI
;'I 'o - u ' t- stoor seq (z'o] r 0 roJ(vus s,orloJ se v 'E'z'I
'o
-
u ' +.ul slooJ)ru
s
LZ
, oI zUISoI SOC
["(t * ry)toc -
:oIloc' oluls :Il:4
"{}ocl3 "I uls :
68 I
o1 lunbo sI uoqenbe Pr
suo4son|- ?uPse1o, su
-
therefore f(x): I torx e R.2
Thus, 1a - -rof a- -2.
2. Letu : x2 * 4x, then u -
(x * 2), -
4 >
168 Solufions fo Testing Quesfions
- -Z satisfies the r
-4 and
y: (u*5)(u+1)+3ut5- r,+9u* 10:x
- -2), therefore ./min : eqz + 9e4) + 10 :3. y- --:', + T =+ U -
L)*' * yx+ (y - n)- o.' xzax*l
When y -
1, then x -
n -
1, so 1 is in the range of y.When y * 1, then above quadratic
equation in x has real roors I ry # 1 in its range. So its
discriminant is non-negative. Therefore
A-y2-4(y-lXy -n) >0=+ 3yr-4(n+t)y*4n S,_
3 ) --rL"rL
3)
therefore (a"l6(n2
- n *
l6(n + 1)2 -a!,m
Lecture \c*'e: ',f,.
-i trer \r\ Z ^l-z,o'hr.hjr 2, so [,4M lmin : 1 Lr *\ow lADlz
-(2a: ] ' -rrhenl
-
nI8 L.(
"*,) + ,(r + ,e *
,(i.")] E=jG*,)+(r+0+ (i.r]
I + DZ+,(NZ-
sl;'(I'o)o
$lBurpJooc oql puu ol ,(sue sl 1l "nol
$l: ,(,- r
pue'? =
lrl u
,?=
c'ttu-roti=
?_
'q'of - z(q +,E : uquouv
:N]
' { Kq, elquqcuor sI Z os 'I -
q -
D JI,(po pue JI sploq ,fir1enbe eql puB
o) =
(rr+rQ*zD)tlunbeur EurtrolloJ eql '9 elduluxg o] m1r-tuls
x-z x-z-
(x){r ,(*-d+I
'rfirlenboul ueoru oq] ,(q '0 < x -
Z eouls
{ loonr,A ruruururiu eqr 'snqr ryuorl^\ elqeurutqo sI ,(q + o)
enlr- etf,L
)*,(r+0+ ,(i*")rlunbe ue^r.r8 eqJ 'serenbs eqt Surleldruoc ,(g
(l t
.Z - "r*(x)_/ 'r*qJ
x''a'\' x -
z - x -
z uoII^\ sploq rfirlunbe eq1
"l
')q
qr(
:1
qr'.:
t*(,plot{
(st\^rrrr( f,t
togJ
(c-SPI(
,(,Ierf,
oJoJoJ
,,,
)'q'o Kve loy.v
_
LZ
spl
r+
,firtT
Z
'r(q + P) slx-r x: r(leuruu' ' :xzQ @ - r)ro
x-Iz+ rQ* rrz -+ +
-7Q z0
e5;-.( I+u)gt
: > ur+ {(t+ u)V-r[t O= (u-aJoJeJeqI'eAltu8eu-uou sr
lrreurjulJ
r,r slooJ IBoJ sur{ r q uorlenbe cr}Jper'd Jo e8uur eq] ur sr I
os
'0 -
(u -
[) + x,( +'oI-
- oI + fu-)e + ,G-t7- : ,? ueq^\ uo>lB] sr I Jo enlB^ urrl't
,(,-_- rC+n) -0I+n6+,fr: g+-t O
puuv_Zn_z(Z-
nbar eql seusrles Z- - o rlluo o, '!t
suo4sen j Suylsel ol sr
'v>D>Iueq^\
'rz,t -
x w I * oZt - uvrlruV.l ?t'gf. %f _ r uer{^\ I + DZ en1eA
luruu1ur-ur s}I se>lgl qcrq/rr 'Z? = lr[ ueqm
69r pedw,{yg [e)neweryery uO seroN onlre'l
-
(ii) (x* y +r)2 > x2*y'*22,3(x*y *z)) y*2y*32
imp,_uur,that
13 .*' + y2 + z2 + z *2y *32 3(1 + {z)@bc)*,therefore
6--- t62(1 + "[Z)r'
where the equalities hold when and only when a -
b -
c. Thus,
5:T/l'maxt62(1 + O),
: The given conditions gives
f(o)_ -11+ al+220
f(x) -
x2-2x-lr( x2 +3
: { x2-2xtl x2-4x-
AC: r/(r+b)2*c,
-
'z; f='r1'I- > x
v+lz-xl'z--D
'0 Z T, + lol-: (I
(g-8) suopsan$ EuPsal
////
a---130
'9I -
_)V'ecue11 '(d urerEurp eq] uI u^\ot{s sB
'og T o) puu 0g T {IVpuu ?8 - ) - q + 0 arctre.Ieq] 'ploq suEts
pnbe oAU eql
'ts : ,(c+q+o)I; )(q+r)i; so)7uls ,q i. olY7 u"'o9 : te
puB'ruc?91 _ )+q+IuJolullrpunb xenuoc B oq
qzZq-grZq-gfi ! t Z
pedwKyg lerrwweryeht uO saroN ernlie-I
: -
i + x)t * r(z + { + r) ) zt +,tZ@ :t+ {Z+ x Z (, + { + x)t,zz *
.,
suoqsen6- ?upse1 ot st
s+ xt-zx II+xZ-zx l _E* zx )
-lf +rl -xZ-zx _ (x){roJ
^roN 'peuplgo sl I ; o ) Z- rlcltl/Y\ ruory
){ puu oZz+lo+rl--(o)/sentE suolllPuoc uenr8 eqJ '1
E(? + I)zstgS
'snql' ) -
q -
u2 uer{^\,(po puB,tZ,t + Dzst
, E(cqo)(gt + r)rteqt pourc]qo sl U '(
li:rtsqns
J;i) (')
-
q : P ueql\ r(1uo puu UE(cqo)gft=
"rzf+rJ3/,'+r!*rc+rqv
v (qe'sr)
*f*zQ
9')q0-
- A sluI'r(1ea,r1cedse"l a' q' o eq J d' g
Z_E-Wr
@){I
(r){I
ruord 'zw) ?9 sl o)gY Jo BorB eqlp eJel{l. ') : Og pue q - 0)'0
: gY tl}lzttO)gV 1ef irroleq (1) uer8elp eq] uI u^\oqs sV i,i(* - s
uer{^\) *zLoE: *z.E -
qzZ * q-rtZ * q-srZ : qzZ * q-trZ : qt * oT tu
ILI
(ss'Er)zD*z)
,(?$z
-
t72 Solutions to Testing Questions
Thus, f (x) > 0 for all x IR, So c6in - -2.
Since a > 0, f (x) takes its minimum value when x
l,f (xo) I -
(3a)b-(a+Dz a2(3a)
Lecture Notes On Mathemi
Since'S x?oos :S x?oog3 t- x?oog A *,0i
anyl0.
l/(1)l -a-b -
f(1)> -b- l/(o)l; and
l,f(xo)l . l/(1)l # a2 + b2 -
ab < 3a(a -
b) +> (a + bt: : r
