<p>Let \(0 \leq \theta \leq 2\pi\) and \(x = |\cos\theta + 1| + |\cos\theta - 1| + |\cos\theta - 2| + |\cos\theta - 3|\), then product of the maximum and minimum values of \(x\) is:</p>
Step-by-Step Solution
Key Concept: Since -1 ≤ cos θ ≤ 1, we can determine the sign of each expression inside absolute values and simplify without absolute value bars. This converts the problem to finding extrema of a piecewise-constant or simple function.
<p><strong>Step 1: Analyze the bounds of cos θ</strong></p><p>Since 0 ≤ θ ≤ 2π, we have -1 ≤ cos θ ≤ 1.</p><p><strong>Step 2: Simplify each absolute value term</strong></p><p>• |cos θ + 1|: Since cos θ ≥ -1, we have cos θ + 1 ≥ 0, so |cos θ + 1| = cos θ + 1</p><p>• |cos θ - 1|: Since cos θ ≤ 1, we have cos θ - 1 ≤ 0, so |cos θ - 1| = 1 - cos θ</p><p>• |cos θ - 2|: Since cos θ ≤ 1 < 2, we have cos θ - 2 < 0, so |cos θ - 2| = 2 - cos θ</p><p>• |cos θ - 3|: Since cos θ ≤ 1 < 3, we have cos θ - 3 < 0, so |cos θ - 3| = 3 - cos θ</p><p><strong>Step 3: Combine all terms</strong></p><p>x = (cos θ + 1) + (1 - cos θ) + (2 - cos θ) + (3 - cos θ)</p><p>x = cos θ + 1 + 1 - cos θ + 2 - cos θ + 3 - cos θ</p><p>x = 7 - 2cos θ</p><p><strong>Step 4: Find maximum and minimum</strong></p><p>Since -1 ≤ cos θ ≤ 1:</p><p>• When cos θ = 1: x = 7 - 2(1) = 5 (minimum)</p><p>• When cos θ = -1: x = 7 - 2(-1) = 9 (maximum)</p><p><strong>Step 5: Calculate the product</strong></p><p>Product = 9 × 5 = 45</p><p>∴ Answer: <strong>45</strong></p>
Correct Answer: 45