<p>There is a cubic polynomial \(f(x)\) with values of \(x\) lying in the interval \([-1, 2]\). Given the condition:</p><p>(i) \(f'''(x) = 24\)</p><p>(ii) An extreme of \(f'(x)\) lies at \(x = \dfrac{-1}{6}\)</p><p>(iii) The coefficient of \(x\) and \(x^0\) in \(f(x)\) are 0 and 6 respectively.</p><p>Find the greatest value of \(f(x)\).</p>
Step-by-Step Solution
Key Concept: Since f'''(x) = 24 (constant), f(x) must be cubic with leading coefficient 4. Use the extremum condition on f'(x) at x = -1/6 (where f''(x) = 0) to find the quadratic coefficient, then apply the coefficient constraints to determine f(x) completely before finding its maximum on [-1, 2].
<p><strong>Step 1: Find the general form of f(x)</strong></p><p>Since f'''(x) = 24 (constant), f(x) is cubic: f(x) = 4x³ + ax² + bx + c</p><p>From condition (iii): coefficient of x⁰ is 6, so c = 6; coefficient of x is 0, so b = 0</p><p>Therefore: f(x) = 4x³ + ax² + 6</p><p><strong>Step 2: Use the extremum condition on f'(x)</strong></p><p>f'(x) = 12x² + 2ax</p><p>f''(x) = 24x + 2a</p><p>An extreme of f'(x) occurs where f''(x) = 0: 24x + 2a = 0</p><p>At x = -1/6: 24(-1/6) + 2a = 0 → -4 + 2a = 0 → a = 2</p><p><strong>Step 3: Write the complete polynomial</strong></p><p>f(x) = 4x³ + 2x² + 6</p><p><strong>Step 4: Find the maximum on [-1, 2]</strong></p><p>f'(x) = 12x² + 4x = 4x(3x + 1)</p><p>Critical points: x = 0 and x = -1/3</p><p>Evaluate at critical points and boundaries:</p><p>f(-1) = 4(-1)³ + 2(-1)² + 6 = -4 + 2 + 6 = 4</p><p>f(-1/3) = 4(-1/27) + 2(1/9) + 6 = -4/27 + 6/27 + 162/27 = 164/27 ≈ 6.07</p><p>f(0) = 6</p><p>f(2) = 4(8) + 2(4) + 6 = 32 + 8 + 6 = 46</p><p><strong>∴ Answer: 46</strong></p>
Correct Answer: 46