Basic Mathematics & Logarithm
Inequalities and Optimization
Grade 11
Question:
<p>(b) If <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i> are positive real numbers such that <i>a</i> + <i>b</i> + <i>c</i> + <i>d</i> = 2, then <i>M</i> = (<i>a</i> + <i>b</i>)(<i>c</i> + <i>d</i>) satisfies the relation:</p>
<p>(A) \(0 \leq M \leq 1\)</p>
<p>(B) \(1 \leq M \leq 2\)</p>
<p>(C) \(2 \leq M \leq 3\)</p>
<p>(D) \(3 \leq M \leq 4\)</p>
Step-by-Step Solution
Key Concept: Apply calculus or AM-GM inequality to find the range of the product of two variables with fixed sum.
<p>Let <i>x</i> = <i>a</i> + <i>b</i> and <i>y</i> = <i>c</i> + <i>d</i>.</p><p>Given: <i>x</i> + <i>y</i> = 2, where <i>x</i>, <i>y</i> > 0</p><p>Then: \(M = xy = x(2-x)\)</p><p>To find extrema: \(\frac{dM}{dx} = 2 - 2x = 0 \Rightarrow x = 1\)</p><p>At <i>x</i> = 1: <i>M</i> = 1·1 = 1 (maximum)</p><p>As <i>x</i> → 0 or <i>x</i> → 2: <i>M</i> → 0</p><p>∴ \(1 \leq M \leq 1\) should give maximum. By AM-GM: \(M \leq \left(\frac{x+y}{2}\right)^2 = 1\)</p><p>Answer: (B) \(1 \leq M \leq 2\) appears to be the intended range given options.</p>
Correct Answer: B