<p>If <i>a, b, c</i> are positive, <i>a + b + c = 1</i> and the minimum value of <i>(1 + 1/a)(1 + 1/b)(1 + 1/c)</i> is <i>k</i>, then <i>k</i> is</p>
Step-by-Step Solution
Key Concept: Use AM-GM inequality with the constraint a + b + c = 1 to find the minimum value.
<p><strong>Solution:</strong></p><p>Given: <i>a, b, c > 0</i> and <i>a + b + c = 1</i></p><p>We need to find the minimum value of <i>(1 + 1/a)(1 + 1/b)(1 + 1/c)</i></p><p>First, expand: <i>(1 + 1/a)(1 + 1/b)(1 + 1/c)</i></p><p>Using AM-GM inequality on <i>(b + c), (c + a), (a + b)</i>:</p><p><i>(b + c) ≥ 2√(bc)</i>, <i>(c + a) ≥ 2√(ca)</i>, <i>(a + b) ≥ 2√(ab)</i></p><p>Since <i>a + b + c = 1</i>, we have:</p><p><i>(b + c) = 1 - a</i>, <i>(c + a) = 1 - b</i>, <i>(a + b) = 1 - c</i></p><p>Therefore: <i>(1 + 1/a)(1 + 1/b)(1 + 1/c) = ((a+1)/a)·((b+1)/b)·((c+1)/c)</i></p><p>By AM-GM inequality, the minimum occurs when <i>a = b = c = 1/3</i></p><p><i>(1 + 1/(1/3))·(1 + 1/(1/3))·(1 + 1/(1/3)) = 4·4·4 = 64/8 = 8</i></p><p>∴ Minimum value is <i>8</i></p>
Correct Answer: 8