Applications of Derivatives
Maxima and Minima of trigonometric functions
Grade 12
Question:
<p>The maximum value of \(f(x) = \cos x(1 + \cos x)\) is greater than its minimum value by:</p>
<p>(a) 1</p>
<p>(b) \(\dfrac{3}{2}\)</p>
<p>(c) 2</p>
<p>(d) \(\dfrac{9}{4}\)</p>
Step-by-Step Solution
Key Concept: Find extrema by taking derivative and setting it to zero, then compute f''(x) to classify critical points. The difference between maximum and minimum values requires evaluating f at critical points found from f'(x) = 0.
<p><strong>Step 1:</strong> Let u = cos x where u ∈ [-1, 1]. Then f(x) = u(1 + u) = u + u²</p><p><strong>Step 2:</strong> Find critical points: g'(u) = 1 + 2u = 0 ⟹ u = -1/2</p><p><strong>Step 3:</strong> Evaluate g(u) at critical point and boundaries:</p><ul><li>At u = -1/2: g(-1/2) = -1/2 + 1/4 = -1/4 (minimum)</li><li>At u = -1: g(-1) = -1 + 1 = 0</li><li>At u = 1: g(1) = 1 + 1 = 2 (maximum)</li></ul><p><strong>Step 4:</strong> Maximum value = 2, Minimum value = -1/4</p><p><strong>Step 5:</strong> Difference = 2 - (-1/4) = 2 + 1/4 = 9/4</p><p>∴ Answer: D (9/4)</p>
Correct Answer: D