39come on , if u want to know how to do in examination, let the roots be 1,2,3,4.
and gven equation is (x-1)(x-2)(x-3)(x-4)
now find a,b,c,d. thats it done.{ in fact this is how i did in exam}
if the equation x4+ax3+bx2+cx+d=0 has four real positive roots then,
a) ac >= 16d
b) b2 >= 36d
c) ac >= 16b
d) b2 >= 36c
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14 Answers
39
24ise kehte hain dhake se question karna.........[3]
but friends yehi cheez exam hall mein kaam aati hai........jis bhi method se ans aaye..bas aana chahiye..........[4]
1waise... solution diya hai ...
koi solution samjha do naa ..... please...
1eureka... am in full support wid you [3]
and am sure anyone who disagrees ... will be purely lying .. sry 'anyone'...
341When they say positive roots inequalities should occur to you.
-a = Σr, -c = d Σ1/r
Hence from AM-GM ac = (-a)(-c) = d Σr Σ1/r ≥ 16d
39bhai ab how can a usless student be compared with a genious teacher?????????????
21Wer did the factor of '16' come frm in prophet sir's soln???
can sum1 xpand....
did it cum frm [x1 + x2 + x3 + x4]/4 = AM or somthin.... pl xplain
341AM-GM is one of the ways of proving that
(x1+x2+x3+x4)(1/x1+1/x2+1/x3+1/x4) ≥ 16
This follows from (x1+x2+x3+x4) ≥ 4 (x1x2x3x4)1/4 and (1/x1+1/x2+1/x3+1/x4) ≥ 4/(x1x2x3x4)1/4
and multiplying the two.
Cauchy Schwarz, AM-HM are two other ways
11x4+ax3+bx2+cx+d=0
a<0,b>0,c<0,d>0
Roots : x1,x2,x3,x4
a= - (x1+x2+x3+x4)
b= (x1x2+x2x3+x3x4+x4x1+x1x3+x2x4)
c= - (x1x2x3+x2x3x4+x3x4x1+x4x1x2)
d= (x1x2x3x4)
If x1=x2=x3=x4=k(let)
a= - 4k
b= 6k2
c= - 4k3
d= k4
ac= 16k4= 16d
b2= 36k4= 36d
So (C) & (D) are incorrect.
If x1=x2=k1(let) & x3=x4=k2(let)
a= -2(k1+k2)
b= (k12+k22+4k1k2)
c= -2k1k2(k1+k2)
d= k12k22
ac= 4k1k2(k1+k2)2 > 16d since (k1+k2)2 > 4k1k2
b2= [ k14+k24+8k1k2(k12+k22) ]+18k12k22 > 36d
since [ k14+k24+ 8k1k2(k12+k22) ] > 18d
Hence (A) & (B) are correct !!
A bit longer but correct approach !!