Basic Mathematics & Logarithm
Inequalities involving means
Grade 11
Question:
<p>The minimum value of \(P = bcx + cay + abz\), when \(xyz = abc\), is</p>
<p>(1) \(3abc\)</p>
<p>(2) \(6abc\)</p>
<p>(3) \(abc\)</p>
<p>(4) \(4abc\)</p>
Step-by-Step Solution
Key Concept: Use the constraint xyz = abc to express variables in terms of each other, then apply AM-GM inequality to find the minimum of the expression.
<p><strong>Step 1:</strong> Given constraint: xyz = abc</p><p><strong>Step 2:</strong> We need to minimize P = bcx + cay + abz subject to xyz = abc.</p><p><strong>Step 3:</strong> Apply weighted AM-GM inequality. For positive numbers with constraint xyz = abc, rewrite:</p><p>P = bcx + cay + abz</p><p><strong>Step 4:</strong> By AM-GM inequality:</p><p>bcx + cay + abz ≥ 3∛(bcx · cay · abz)</p><p><strong>Step 5:</strong> Calculate the product under the cube root:</p><p>bcx · cay · abz = (abc)² · xyz = (abc)² · abc = (abc)³</p><p><strong>Step 6:</strong> Therefore:</p><p>P ≥ 3∛((abc)³) = 3abc</p><p><strong>Step 7:</strong> Equality holds in AM-GM when bcx = cay = abz</p><p>Combined with xyz = abc, this gives x = a, y = b, z = c</p><p>∴ <strong>Minimum value of P = 3abc</strong></p>
Correct Answer: A