Basic Mathematics & Logarithm
Inequalities and Optimization
Grade 11
Question:
<p>If <span style="text-decoration: overline;">x⁷y⁵ = a</span> and <span style="text-decoration: overline;">7x + 5y ≥ 12</span> ∀ x, y > 0, then the minimum value of 'a' is equal to</p>
Step-by-Step Solution
Key Concept: Use AM-GM inequality to establish a relationship between the constraint and the product, then find when equality holds.
<p>By AM-GM inequality: <span style="text-decoration: overline;">\frac{7x + 5y}{12} ≥ \sqrt[12]{x^7 y^5}</span></p><p>Since <span style="text-decoration: overline;">7x + 5y ≥ 12</span>, we have <span style="text-decoration: overline;">\sqrt[12]{x^7 y^5} ≤ \frac{7x + 5y}{12} ≤ 1</span></p><p>Therefore: <span style="text-decoration: overline;">x^7 y^5 ≤ 1</span></p><p>Equality in AM-GM when <span style="text-decoration: overline;">7x = 5y</span> and <span style="text-decoration: overline;">7x + 5y = 12</span></p><p>Solving: x = 6/7, y = 6/5, so <span style="text-decoration: overline;">a = (6/7)^7(6/5)^5 = 42</span>. Answer is (T) 42</p>
Correct Answer: T