<p>The maximum value of \(f(x) = \cos x(1 + \cos x)\) is greater than its minimum value by:</p>
Step-by-Step Solution
Key Concept: Find the extreme values of f(x) = cos x(1 + cos x) by substituting t = cos x where t ∈ [-1,1], then analyze the quadratic g(t) = t(1+t) = t² + t.
<p><strong>Step 1:</strong> Let t = cos x, where t ∈ [-1, 1]</p><p>Then f(x) = t(1 + t) = t² + t</p><p><strong>Step 2:</strong> Define g(t) = t² + t on [-1, 1]. Find critical points:</p><p>g'(t) = 2t + 1 = 0 ⟹ t = -1/2</p><p><strong>Step 3:</strong> Evaluate g(t) at critical point and endpoints:</p><p>• g(-1/2) = 1/4 - 1/2 = -1/4</p><p>• g(-1) = 1 - 1 = 0</p><p>• g(1) = 1 + 1 = 2</p><p><strong>Step 4:</strong> Maximum value = 2 (at t = 1, i.e., cos x = 1)</p><p>Minimum value = -1/4 (at t = -1/2, i.e., cos x = -1/2)</p><p><strong>Step 5:</strong> Difference = 2 - (-1/4) = 2 + 1/4 = 9/4</p><p>∴ Answer: <strong>9/4</strong></p>
Correct Answer: D