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If x, y, z are real numbers and given that xyz is equal to product of a, (x + y), (y + z) and (z + x). Where a is any real number. Then find minimum possible value of a.
Give this question a try then check your answer with solution.
Given equation: xyz = a(x + y)(y + z)(z + x).
We have to find amin.
We know that AM ≥ GM
For x, y:
\[\frac{x+y}{2}\geq\sqrt{xy}----(1)\]
For y, z:
\[\frac{y+z}{2}\geq\sqrt{yz}----(2)\]
For z, x:
\[\frac{z+x}{2}\geq\sqrt{zx}----(3)\]
Multiply LHS and RHS of (1), (2), (3) and equate them:
\[\left(\frac{x+y}{2}\right)\left(\frac{y+z}{2}\right)\left(\frac{z+x}{2}\right)\geq\sqrt{(xy)(yz)(zx)}\]
\[\Rightarrow\frac{(x+y)(y+z)(z+x)}{8}\geq xyz\]
\[\Rightarrow xyz\leq\frac{(x+y)(y+z)(z+x)}{8}\]
Here we can clearly observe that value of a cannot less than 1/8.
⇒ amin = ⅛
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