<p>Let <em>M</em> and <em>m</em> be respectively the absolute maximum and the absolute minimum values of the function, \(f(x) = 2x^3 - 9x^2 + 12x + 5\) in the interval [0, 3]. Then <em>M</em> − <em>m</em> is equal to</p>
Step-by-Step Solution
Key Concept: Find critical points by setting f'(x) = 0, then evaluate f at critical points and endpoints to identify absolute extrema on the closed interval [0,3].
<p><strong>Step 1:</strong> Find the derivative and critical points.</p><p>f(x) = 2x³ - 9x² + 12x + 5</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</p><p><strong>Step 2:</strong> Evaluate f at critical points and endpoints.</p><p>f(0) = 5</p><p>f(1) = 2(1) - 9(1) + 12(1) + 5 = 2 - 9 + 12 + 5 = 10</p><p>f(2) = 2(8) - 9(4) + 12(2) + 5 = 16 - 36 + 24 + 5 = 9</p><p>f(3) = 2(27) - 9(9) + 12(3) + 5 = 54 - 81 + 36 + 5 = 14</p><p><strong>Step 3:</strong> Identify absolute maximum and minimum.</p><p>M = max{5, 10, 9, 14} = 14</p><p>m = min{5, 10, 9, 14} = 5</p><p><strong>Step 4:</strong> Calculate M - m.</p><p>M - m = 14 - 5 = 9</p><p>∴ Answer: B (9)</p>
Correct Answer: B