Question

If x + y + z = 1 and x,y,z are positive numbers such that $(1-x)(1-y)(1-z)\ge kxyz$, then exact value of k is equal to

• A. 2
• B. 4
• C. 8
• D. 16

Answer 1

The correct option is C 8

Since $A.M.\ge G.M.\therefore \frac{x+y}{2}\ge \sqrt{xy}...\left(i\right)\frac{y+z}{2}\ge \sqrt{yz}...\left(ii\right)\frac{z+x}{2}\ge \sqrt{zx}...\left(iii\right)Multiplying\left(i\right),\left(ii\right)\left(iii\right)weget\frac{(x+y)(y+z)(z+x)}{8}\ge xyz$
or$(1-x)(1-y)(1-z)\ge 8xyz$