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Question

If n>m, y>n, z>r,(x,y,z>0) such that āˆ£āˆ£ āˆ£āˆ£xnrmyrmnzāˆ£āˆ£ āˆ£āˆ£=0 then greatest value of 27 xyz(xāˆ’m)(yāˆ’n)(zāˆ’r) is

A
-1
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B
8
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C
1
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D
2
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Solution

The correct option is B 8
Given that,
∣ ∣xnrmyrmnz∣ ∣=0(R1R1R2R2R2R3)
∣ ∣xmny00ynrzmnz∣ ∣=0
(xm)[z(yn)n(rz)]+m[(ny)(rz)]=0
zzr+nyn+mxm=0
zzr+nyn+1+mxm+1=2
zzr+n+ynyn+m+xmxm=2
zzr+yyn+xxm=2
Applying AM -GM Inequality
xxm+yyn+zzr3(xyz(xm)(yn)(zr))13
xyz(xm)(yn)(zr)827
So 27xyz(xm)(yn)(zr)8

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