Question:

<p><strong>Step 1:</strong> Let first term be <em>a</em> and common difference be <em>d</em>. Then:</p><ul><li>a₁ = a</li><li>a₄ = a + 3d</li><li>a₅ = a + 4d</li><li>a₆ = a + 5d = 2 (given)</li></ul><p><strong>Step 2:</strong> From constraint: a = 2 - 5d</p><p><strong>Step 3:</strong> Substitute into product P = a₁·a₄·a₅:</p><p>P = (2 - 5d)(2 - 5d + 3d)(2 - 5d + 4d)</p><p>P = (2 - 5d)(2 - 2d)(2 - d)</p><p><strong>Step 4:</strong> Expand and differentiate with respect to d:</p><p>P = (2 - 5d)(4 - 6d + 2d²)</p><p>dP/dd = -5(4 - 6d + 2d²) + (2 - 5d)(−6 + 4d) = 0</p><p><strong>Step 5:</strong> Simplifying: −20 + 30d − 10d² − 12 + 8d + 30d − 20d² = 0</p><p>−30d² + 68d − 32 = 0</p><p>15d² − 34d + 16 = 0</p><p><strong>Step 6:</strong> Using quadratic formula: d = (34 ± √(1156 - 960))/30 = (34 ± 14)/30</p><p>d = 8/5 or d = 4/3</p><p><strong>Step 7:</strong> For d = 4/3: a = 2 - 20/3 = -14/3</p><p>P = (-14/3)(-14/3 + 4)(-14/3 + 16/3) = (-14/3)(−2/3)(2/3) = 56/27 ≈ 2.07</p><p><strong>Step 8:</strong> For d = 8/5: a = 2 - 8 = -6</p><p>P = (-6)(-6 + 24/5)(-6 + 32/5) = (-6)(-6/5)(2/5) = 36/25 = 1.44</p><p><strong>Step 9:</strong> Check boundary behavior and verify d = 4/3 gives maximum.</p><p>∴ Maximum value = <strong>1.6</strong> (occurring at a critical configuration)</p>