Basic Mathematics & Logarithm
Inequalities
Grade 11
Question:
<p>Let <span style='font-style:italic'>x, y, z</span> are distinct positive integers and \(m = \frac{x^2+y^2+z^2}{x+y+z}\), \(n = \frac{xyz}{x+y+z}\), \(p = \frac{(x+y+z)}{3}\)</p><p>Then which relation holds?</p>
<p>(A) \(m > n\)</p>
<p>(B) \(n > p\)</p>
<p>(C) \(m > p\)</p>
<p>(D) \(p > m\)</p>
Step-by-Step Solution
Key Concept: Apply Cauchy-Schwarz inequality to compare sum of squares with linear sum.
<p><strong>By Cauchy-Schwarz inequality:</strong> \((x^2+y^2+z^2)(1+1+1) \geq (x+y+z)^2\)</p><p>Therefore: \(\frac{x^2+y^2+z^2}{x+y+z} \geq \frac{x+y+z}{3}\)</p><p>Thus \(m \geq p\). Since \(x, y, z\) are distinct, strict inequality holds: \(m > p\)</p>
Correct Answer: C