prove it (imp 4 jee)

any condition on x andy given????

This is d famous Nesbitts Inequality ....

its got some 25 proofs ...
[1]

aage mirka ji

please elaborate a bit more in #3
i am weak in HM inequality

stillwaiting for 24 [3]

lets move on Proof No 2

we have

x²/xz+xy +y²/xy+yz + z²/xz+yz ≥3/2

Applying Cauchy we get

(x+y+z)²/2(x+y+z)≥3/2

x²+y²+z²â‰¥xy+yz+xz
1/2[(x-y)²+(y-z)²+(x-z)²]≥0

hence proved

by symmetry minima if exists is at x=y=z

so put x=y=z u get ans≥3/2

simplest proof -

and if u want the 25 proofs, here it is (from Kaymant sir's page) :

http://kaymant.googlepages.com/HowmanyproofsoftheNesbittsInequanlit.pdf

awesome post mirka
thanx 4 the link
now its party (inequality )time