<p>If \(a, b, c\) and \(d\) are four positive real numbers such that \(abcd = 1\), the minimum value of \((1 + a)(1 + b)(1 + c)(1 + d)\) is</p>
Step-by-Step Solution
Key Concept: Use the AM-GM inequality applied strategically to the expanded product, recognizing that the constraint abcd = 1 creates a symmetric structure where equality occurs when a = b = c = d = 1.
<p><strong>Step 1: Apply AM-GM to pairs of factors</strong></p><p>For positive reals, by AM-GM inequality: (1+a)(1+b) ≥ (√(ab) + √(ab))² = 4√(ab) is not the optimal form.</p><p><strong>Step 2: Use direct AM-GM on the product form</strong></p><p>By AM-GM inequality applied to the four terms (1+a), (1+b), (1+c), (1+d):</p><p>$$\frac{(1+a)+(1+b)+(1+c)+(1+d)}{4} \geq \sqrt[4]{(1+a)(1+b)(1+c)(1+d)}$$</p><p><strong>Step 3: Alternative approach - expand and apply AM-GM strategically</strong></p><p>Expand: $(1+a)(1+b)(1+c)(1+d) = 1 + (a+b+c+d) + (ab+ac+ad+bc+bd+cd) + (abc+abd+acd+bcd) + abcd$</p><p>Since $abcd = 1$, the last term equals 1.</p><p><strong>Step 4: Apply AM-GM to minimize</strong></p><p>By AM-GM: $a+b+c+d \geq 4\sqrt[4]{abcd} = 4\sqrt[4]{1} = 4$</p><p>Similarly, by AM-GM on products: $abc+abd+acd+bcd \geq 4\sqrt[4]{(abc)(abd)(acd)(bcd)} = 4\sqrt[4]{a^3b^3c^3d^3} = 4(abcd)^{3/4} = 4$</p><p>And $ab+ac+ad+bc+bd+cd \geq 6\sqrt[6]{(ab)(ac)(ad)(bc)(bd)(cd)} = 6(abcd)^{2/3} \cdot \sqrt[6]{abcd} = 6(abcd) = 6$</p><p><strong>Step 5: Find minimum value</strong></p><p>Therefore: $(1+a)(1+b)(1+c)(1+d) \geq 1 + 4 + 6 + 4 + 1 = 16$</p><p><strong>Step 6: Verify equality condition</strong></p><p>Equality holds when $a = b = c = d$ and $abcd = 1$, giving $a = b = c = d = 1$.</p><p>Check: $(1+1)(1+1)(1+1)(1+1) = 2 \times 2 \times 2 \times 2 = 16$ ✓</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C