inequality problems

8/13/2019 Inequality problems

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Inequality Problems in QQAD so far

Type: Manipulating Expressions

Which of the following statements about the functions

A(x, y, z) = x + 3y - 4xy - yz + zx an! "(x,y,z) = #x-#x-y## + #y-#y-z## + #z-#z-x## is necessa$ily t$ue%

(a) "(x,y,z) &= #x-y# + #y-z# + #z-x# (b) ("(x,y,z))' A(x,y,z) fo$ x y z

(c) A(x,y,z) -* (!) one of the aboe

ince all the te$ms in "(x,y,z) a$e symmet$ic we assume without the loss of anygene$ality that x = y = z

.hus, "(x,y,z) = #x-#x-y## + #y-#y-z## + #z-#z-x## = #y# + #z# + #z-x##x-y# + #y-z# + #z-x# = x-y + y-z + x-z = (x-z)

/utting x = 0, y=0, z= 0 we see that choice (a) is not t$ue

x + 3y - 4xy - yz + zx = (x - y + z)' -y' -z' + yz= A(x,y,z) = (x - y + z)' -(y-z)'

= A(x,y,z) = (x - 3y + z)(x-y)

1et x y z an! x z, then "(x,y,z) = x+y-z, if ("(x,y,z))' A(x,y,z) =("(x,y,z))' - A(x,y,z) , soling we get y(3x-y) + x(3y-z) which is

1et x y z an! x & z, then "(x,y,z) = y+z+(z-x) = 3z +y -x if ("(x,y,z))' A(x,y,z) = ("(x,y,z))' - A(x,y,z) , soling we get (3z +y -x)' - (x - 3y + z)(x-y) = 2z' - y' +yz +xy -zx

= z(2z-x+y) + x(x-y) , which we can see always is ence, (b) is the $ighto/tion

/ut x=0, y = , z = -4 we get A(x, y, z) = -5hence (c) is also not t$ue


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4/13

Type: #actori(ation base& on !AT )**+ question"

1et x = (n'4 +


5/13

(a' + b')D(c' + !') = (ac + b!)'2D< = ((x-3)' + (y+)')D(3' + 4') = (3(x-3) + 4(y+))'= 2D< = (3(x-3) + 4(y+))'

= 3x + 4y &= 0*

ence, choice (!) is the co$$ect answe$

.his was base! on cauchy-schwa$zHs ineFuality@

Type: !auc'y%-c'.ar(

6f a, b, c, ! a$e $eal numbe$s with a' + b' + c' + !' = 0, then what is themaximum alue of a + 3b + *c + 4!%

(a) 4 (b)


6/13

+3 & D3 = * so this means we shoul! 7ust !ec$ease the numbe$ 3 untill it getsto 04 an! inc$ease all the othe$ numbe$s by [email protected] we get the solution D3D@@@D0D0D03D04 o$ sho$tly 04;4

ence, c'oice b"is the $ight answe$

Type: !alculus an& Equations QQAD )**7 an& later appeare& in 8AT)**5"

6f a, b, c a$e $eal numbe$s such that a & b & c an! a + b + c = *, ab + bc + ca = 2,then which among the following is !efinitely t$ue%

(a) & a & 0 (b) 0 & b & 3 (c) 3 & c & 4 (!) All of them

1et t = abc@ .hen a, b, c a$e the $oots (= ze$os) of f(x) = x'3 - *x' + 2x - t = @We hae fI(x) = 3(x - 0)(x - 3) so f(3) & & f(0)@ ence a & 0 & b & 3 & c@ince f(0) = 4 - t , t = abc & 4, an! since f(3) = -t & , abc @As b, c a$e /ositie, a @Also, f(x) is st$ictly inc$easing on (3, +oo) an! f(4) = f(0) = 4 - / so c & 4@

ence, c'oice &"is the $ight answe$

Type: Anot'er variant of !auc'y%-c'.ar(

.he sum of fie $eal numbe$s is an! the sum of thei$ sFua$es is 0*@ What is thela$gest /ossible alue fo$ one of the numbe$s%

(a) 0*;< (b) (c)


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8/13

Type: or& problem for AM$GM in equations

A//le ha! to stu!y 9athematics f$om his teache$ B$ange@ As usual, A//le wasslee/ing in the class@ 6mme!iately he hea$! a thun!e$ing oice@ MAll $oots of this

eFuation a$e $eal as well as /ositie in natu$eM@ A//le woNe u/ f$om !ee/slumbe$@ e hu$$ie! to co/y the 0 !eg$ee eFuation w$itten on the boa$! butcoul! co/y only the fi$st two te$ms w$itten on the blacNboa$! befo$e B$ange si$wi/e! it all@ A//le howee$ $emembe$e! that the constant te$m was @ e note!!own the eFuation asx'0 - x'2OOO@ + = @ A//le was e$y su$e that ifsomeone woul! tell him the sum of all the coefficients of all the /owe$s of x in theeFuation, he woul! sole it anyhow@ e asNe! 9ango about the same@ .he answe$which 9ango co$$ectly gae was

(a) -04 (b) (c) 04 (!) 9ango himself was confuse! (e) none ofthe fo$egoing

olutionJ

1et the $oots be a0, a,O@@, a0 then a0 + a + O@ + a0 = 0 , an! (a0@aO@a0)=0ow since all the $oots a$e $eal an! /ositie in natu$eWe can say that (a0+a+O@+a0);0=(a0@aO@a0)'0;0o$ $eal numbe$s, A9 = C9 but he$e we fin! that A9 = C9 hence a0 = a = O= a0 = 0o ou$ eFuation actually is (x-0)'0= an! then the sum of all the coefficients ofall the /owe$s of x in the eFuation is @

ence, choice (b) is the $ight answe$

Type: Pigeon ;ole in Inequality

.he minimum /ossible alue of the la$gest of ab, 0-a-b+ab, an! a+b-ab if &= a&= b &=0 is

(a) 4;2 (b)0;2 (c)


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11/13

A" 016 ," ) D" )1? D" 01+ E" )1@

We nee! to fin! the g$eatest alue of /F(/'+F') gien /' + /F + F' = 0, but

/' + F' = /F = / = F >/' + F' = /F !oesnHt hae any $eal solution set?@ence, choice (e) is the co$$ect answe$

Type: Max of Min< or Min of Max

a, b, c, !, e, f, g a$e non-negatie such that a+b+c+!+e+f+g = 0@ .hen theminimum alue of max(a+b+c, b+c+!, c+!+e, !+e+f, e+f+g) is

a" 01+ b" +14 c" 0 &" * e" none of t'e foregoing

ince the$e a$e 3n- te$ms, /ut a=!=f=0;3

Type: Evaluating a expression in anot'er form

6f th$ee /ositie $eal numbe$s x, y, z satisfy y-x = z-y an! xyz = 2, then theminimum /ossible alue of y is

(0) 3 () 3;3 (3) 30;3 (4) 0 (


12/13

ole you$self, Answe$ is o/tion (3)

Type: Inequality problem t'roug' substitution an& symmetry

1et x an! y be /ositie $eal numbe$s such that x'3 + y'3 = 4x'@ A is themaximum alue of x + y@1et a, b, c be $eal such that a+b+c = < an! ab + bc + ca = 3@ " is the la$gest/ossible alue of c@

.hen A + " lies in the $ange

(0) >5, ) () >, 2) (3) >2, 0) (4) >0, 00) (


13/13

Type: Inequality problem of Minimi(ing sum .'en pro&uct isconstant

1et /, F be /ositie $eal numbe$s such that /'3@F = [email protected] min alue of /+F is

/ + F = / + ;/'3 = /;3 + /;3 + /;3 + ;/'3 = 4(;5)'0;4 >f$om A9KC9?

Putting it as a formula:

to find the min value of p(x)=x-a|+|x-b|+|x-c|+|x-d+....

1) hen no. of terms are odd! min. value of P(x) ould be atx=median(a!b!c!d!...).

") hen no. of terms are even! min p(x)= d-a|+|b-c| i.e (|nth term- 1st term| + n-1th term- "nd term + ---- )! here a

inequality problems

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