Sets, Relations & Functions
Grade 11
Question:
<p>If \(2^{1+x}+2^{1-x}\), \(f(x)\), \(3^x+3^{-x}\) are in AP, find min \(f(x)\).</p>
Step-by-Step Solution
<div class="solution"><p>f(x)=(2^{1+x}+2^{1-x}+3^x+3^{-x})/2. By AM-GM: 2^{1+x}+2^{1-x}\ge4 and 3^x+3^{-x}\ge2. Min f=3 at x=0.</p><div class="trap-box"><strong>Trap:</strong> Don't minimize by calculus before seeing the AM-GM form.<div class="key-concept"><strong>Key Concept:</strong> aˣ+a⁻ˣ\ge2 (AM-GM)
Correct Answer: 3