Basic Mathematics & Logarithm
Inequalities and Optimization
Grade 11
Question:
<p><strong>(B)</strong> For positive numbers <math>a, b, c</math> the minimum value of <math>\frac{a(b^2 + c^2) + b(c^2 + a^2) + c(a^2 + b^2)}{abc}</math> is equal to</p>
<p>(P) 1</p>
<p>(Q) 2</p>
<p>(R) 3</p>
<p>(S) 4</p>
<p>(T) 6</p>
Step-by-Step Solution
Key Concept: Use AM-GM inequality to find minimum values of symmetric expressions in multiple variables.
<p><strong>Solution:</strong> Rewrite the expression as:</p><p><math>\frac{a(b^2 + c^2) + b(c^2 + a^2) + c(a^2 + b^2)}{abc} = \frac{b}{c} + \frac{c}{b} + \frac{c}{a} + \frac{a}{c} + \frac{a}{b} + \frac{b}{a}</math></p><p>By AM-GM inequality:</p><p><math>\frac{b}{c} + \frac{c}{b} \geq 2, \quad \frac{c}{a} + \frac{a}{c} \geq 2, \quad \frac{a}{b} + \frac{b}{a} \geq 2</math></p><p>Therefore, the minimum value is <math>2 + 2 + 2 = 6</math>, achieved when <math>a = b = c</math>.</p>
Correct Answer: T