2025 · Q2
Consider the following statements for a positive real number x:
- Ix + 1/x > 1
- II(x + 1/x)^2 > 2
- III(x + 1/x)^4 > 9 Which of the statements given above are correct?
Elementary MathematicsAlgebra
- AI and II only
- BII and III only
- CI and III only
- DI, II and III
Explanation
For x>0, AM-GM gives x+1/x>=2. Hence x+1/x>1 (I true), (x+1/x)^2>=4>2 (II true), and (x+1/x)^4>=16>9 (III true). All three statements hold, so the answer is I, II and III. The trap is doubting the higher powers, but monotonicity of t^2 and t^4 for t>=2 settles it.