<p><strong>768.</strong> There is a cubic polynomial \(f(x)\) with values of \(x\) lying in the interval \([-1, 2]\). Given the condition:<br>(i) \(f'''(x) = 24\)<br>(ii) An extreme of \(f'(x)\) lies at \(x = \dfrac{-1}{6}\)<br>(iii) The coefficient of \(x\) and \(x^0\) in \(f(x)\) are 0 and 6 respectively.<br>Find the greatest value of \(f(x)\).</p>
Step-by-Step Solution
Key Concept: For a cubic polynomial, f'''(x) = 24 gives the leading coefficient (a=4), then use f''(x) having an extremum at x=-1/6 to find remaining coefficients, and apply the boundary/critical point analysis on [-1,2].
<p><strong>Step 1: Find leading coefficient</strong></p><p>Let f(x) = ax³ + bx² + cx + d. Since f'''(x) = 24, we have 6a = 24, so <strong>a = 4</strong>.</p><p><strong>Step 2: Use condition (iii)</strong></p><p>Coefficient of x⁰ (constant term) is d = 6, and coefficient of x is c = 0.</p><p>So f(x) = 4x³ + bx² + 6.</p><p><strong>Step 3: Use extremum condition</strong></p><p>f'(x) = 12x² + 2bx, so f''(x) = 24x + 2b.</p><p>f'(x) has an extreme at x = -1/6, meaning f''(-1/6) = 0:</p><p>24(-1/6) + 2b = 0 ⟹ -4 + 2b = 0 ⟹ <strong>b = 2</strong>.</p><p><strong>Step 4: Complete polynomial</strong></p><p>f(x) = 4x³ + 2x² + 6</p><p><strong>Step 5: Find maximum on [-1, 2]</strong></p><p>f'(x) = 12x² + 4x = 4x(3x + 1) = 0 ⟹ x = 0 or x = -1/3</p><p>Evaluate at critical points and boundaries:</p><p>• f(-1) = 4(-1) + 2(1) + 6 = 4</p><p>• f(-1/3) = 4(-1/27) + 2(1/9) + 6 = -4/27 + 6/27 + 6 = 2/27 + 6 ≈ 6.074</p><p>• f(0) = 6</p><p>• f(2) = 4(8) + 2(4) + 6 = 32 + 8 + 6 = <strong>46</strong></p><p>∴ Answer: <strong>46</strong></p>
Correct Answer: 46