<p>Given: <br>
\(f(x) = 2x^3 - 9x^2 + 12x + 5\) on \([0, 3]\)<br>
Find \(M - m\) where \(M\) is the maximum value and \(m\) is the minimum value of \(f(x)\) on \([0,3]\).</p>
Step-by-Step Solution
Key Concept: To find the absolute maximum and minimum of a continuous function on a closed interval, evaluate f at all critical points (where f'(x)=0) and at both endpoints, then compare all values.
<p><strong>Step 1:</strong> Find critical points by computing f'(x)</p><p>f'(x) = 6x² - 18x + 12 = 6(x² - 3x + 2) = 6(x-1)(x-2)</p><p>Critical points: x = 1 and x = 2 (both lie in [0,3] ✓)</p><p><strong>Step 2:</strong> Evaluate f(x) at critical points and endpoints</p><p>f(0) = 2(0)³ - 9(0)² + 12(0) + 5 = <strong>5</strong></p><p>f(1) = 2(1) - 9(1) + 12(1) + 5 = 2 - 9 + 12 + 5 = <strong>10</strong></p><p>f(2) = 2(8) - 9(4) + 12(2) + 5 = 16 - 36 + 24 + 5 = <strong>9</strong></p><p>f(3) = 2(27) - 9(9) + 12(3) + 5 = 54 - 81 + 36 + 5 = <strong>14</strong></p><p><strong>Step 3:</strong> Compare all values</p><p>Values: {5, 10, 9, 14}</p><p>Maximum M = 14 (at x = 3)</p><p>Minimum m = 5 (at x = 0)</p><p><strong>∴ M - m = 14 - 5 = 9</strong></p>
Correct Answer: 9