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?
Show Hint
- \(y>z>x\)
- \(z>x>y\)
- \(x<z>y\)
- \(x<y<z\)
The Correct Option is D
Solution and Explanation
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. Step 2: Determine Order:
This suggests \(x\) is the smallest variable and \(z\) is the largest.
\(y\) lies in between.
Order: \(x<y<z\).
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Questions Asked in SRCC GBO exam
Read the given passage and answer the questions.
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