Question

If x, y, z are distinct positive integers then value of expression (x + y) (y + z) (z + x) is

• A. is equal to 8xyz
• B. is greater than 8xyz
• C. is smaller than 8xyz
• D. is equal to 4xyz

Answer 1

The correct option is B is greater than 8xyz


$\mathrm{Since}x,y,z\mathrm{are}\mathrm{distinct}\mathrm{positive}\mathrm{integers}.A.M.>G.M.(\mathrm{Equality}\mathrm{doesn}\text{'}t\mathrm{hold}\mathrm{because}\mathrm{the}\mathrm{numbers}\mathrm{are}\mathrm{distinct})\Rightarrow \frac{x+y}{2}>\sqrt{\mathrm{xy}},\frac{y+z}{2}>\sqrt{\mathrm{yz}},\frac{z+x}{2}>\sqrt{\mathrm{zx}}\mathrm{Multiplying}\mathrm{the}\mathrm{three}\mathrm{inequalities},\mathrm{we}\mathrm{get},\Rightarrow \left(\frac{x+y}{2}\right)\left(\frac{y+z}{2}\right)\left(\frac{z+x}{2}\right)>\sqrt{\mathrm{xy}\times \mathrm{yz}\times \mathrm{zx}}\mathrm{Note}:\mathrm{When}\mathrm{both}\mathrm{the}\mathrm{sides}\mathrm{of}\mathrm{inequalities}\mathrm{are}\mathrm{positive}\mathrm{and}\mathrm{we}\mathrm{multiply}\mathrm{the}\mathrm{inequalities},\mathrm{then}\mathrm{the}\mathrm{inequality}\mathrm{sign}\mathrm{remain}\mathrm{the}\mathrm{same}.\mathrm{In}\mathrm{general},\mathrm{if}\mathrm{we}\mathrm{don}\text{'}t\mathrm{know}\mathrm{the}\mathrm{signs}\mathrm{of}\mathrm{both}\mathrm{the}\mathrm{sides}\mathrm{of}\mathrm{inequalities},\mathrm{then}\mathrm{we}\mathrm{can}\text{'}t\mathrm{multuply}\mathrm{them}.\Rightarrow \left(x+y\right)\left(y+z\right)\left(z+x\right)>8\mathrm{xyz}$