Basic Mathematics & Logarithm
Inequalities involving means
Grade 11
Question:
<p>If \(y = 3^{x-1} + 3^{-x-1}\), then the least value of \(y\) is</p>
<p>(1) 2</p>
<p>(2) 6</p>
<p>(3) 2/3</p>
<p>(4) 3/2</p>
Step-by-Step Solution
Key Concept: Recognize that y = 3^(x-1) + 3^(-x-1) = (1/3)·3^x + (1/3)·3^(-x) is a sum of exponential and its reciprocal. Apply AM-GM inequality to find the minimum value when both terms are equal.
<p><strong>Step 1:</strong> Rewrite y by factoring constants:</p><p>y = 3^(x-1) + 3^(-x-1) = (1/3)·3^x + (1/3)·3^(-x) = (1/3)(3^x + 3^(-x))</p><p><strong>Step 2:</strong> Apply AM-GM inequality to (3^x + 3^(-x)):</p><p>3^x + 3^(-x) ≥ 2√(3^x · 3^(-x)) = 2√(3^0) = 2√1 = 2</p><p><strong>Step 3:</strong> Equality holds when 3^x = 3^(-x), which gives x = 0</p><p><strong>Step 4:</strong> Substitute back into original expression:</p><p>y_min = (1/3) · 2 = 2/3</p><p>∴ Answer: C (The least value of y is <strong>2/3</strong>)</p>
Correct Answer: C