JEE Advanced Maths: Inequalities / Algebra questions with solutions

Q1. Let x, y, z be positive real numbers such that x²*y*z + 4*x*y*z + 4*y*z = 81. The minimum value of 2x + y + z:

1. is greater than 27
2. cannot be attained
3. is less than 9
4. lies between 9 and 27

Answer: is less than 9

The constraint factors to y*z*(x+2)² = 81. Setting u = x+2 and using AM-GM on y+z >= 2*sqrt(yz) = 18/u, the expression 2x+y+z = 2u-4+y+z >= 2u+18/u-4, minimised at u=3 giving 8.