Question

If x + y + z = 1 and x, y, z are positive numbers such that $(1-x)(1-y)(1-z)\ge kxyz$, then k =

• A. 8
• B. 6
• C. 9
• D. 4

Answer 1

The correct option is A 8

Since A.M. $\ge$ G.M.
$\therefore \frac{y+z}{2}\ge \sqrt{yz}$ ...(1)
$\frac{z+x}{2}\ge \sqrt{zx}$ ....(2)
$\frac{y+x}{2}\ge \sqrt{yx}$ ...(3)
Multiplying (1), (2) and (3), we get
$\frac{(y+z)(z+x)(x+y)}{8}\ge xyz$
or $(1-x)(1-y)(1-z)\ge 8xyz$ $(\because x+y+z=1)$