Question

If \(\frac{x^3}{z^2}<\frac{x^3+y^3+z^3}{x^2+y^2+z^2}<\frac{z^3}{x^2}\); \(x, y, z\) are positive real numbers, then which of the following options always ensure the given inequality to be true?

• A. \(y>z>x\)
• B. \(z>x>y\)
• C. \(x<z>y\)
• D. \(x<y<z\)

Hint:

Substitute simple values like \(x=1, y=2, z=3\) to verify the inequality holds.

Answer 1

The Correct Option is D

Strategy:

The middle term is an average-like quantity.
The lower bound is \(\frac{x^3}{z^2}\). For this to be the smallest value, \(x\) should be small and \(z\) should be large.
The upper bound is \(\frac{z^3}{x^2}\). For this to be the largest value, \(z\) should be large and \(x\) should be small.
This suggests \(x\) is the smallest variable and \(z\) is the largest.
\(y\) lies in between.
Order: \(x<y<z\).