<p>The number of points of intersection of \(2y = 1\) and \(y = \cos x\) in \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\) is</p>
Step-by-Step Solution
Key Concept: We need to find where the horizontal line y = 1/2 intersects the cosine curve y = cos x on the interval [-π/2, π/2]. This requires analyzing the behavior of cos x and solving cos x = 1/2 within the given domain.
<p><strong>Step 1:</strong> Rewrite the first equation. We have 2y = 1, which gives y = 1/2.</p><p><strong>Step 2:</strong> We need to find intersection points of y = 1/2 and y = cos x for x ∈ [-π/2, π/2].</p><p><strong>Step 3:</strong> This requires solving cos x = 1/2 in the interval [-π/2, π/2].</p><p><strong>Step 4:</strong> Recall that cos x = 1/2 has solutions at x = ±π/3 (in the principal range).</p><p><strong>Step 5:</strong> Check if both solutions lie in [-π/2, π/2]:<br/>- x = π/3 ≈ 1.047, and π/2 ≈ 1.571, so π/3 is in the interval ✓<br/>- x = -π/3 ≈ -1.047, and -π/2 ≈ -1.571, so -π/3 is in the interval ✓</p><p><strong>Step 6:</strong> Verify using properties of cosine: Since cos x is an even function and symmetric about the y-axis, if x = π/3 is a solution, then x = -π/3 is also a solution.</p><p><strong>Step 7:</strong> At the boundaries:<br/>- At x = π/2: cos(π/2) = 0 ≠ 1/2<br/>- At x = -π/2: cos(-π/2) = 0 ≠ 1/2<br/>So the boundary points do not contribute additional intersections.</p><p><strong>Step 8:</strong> Between the solutions, cos x varies continuously from 0 at x = -π/2, increases to 1 at x = 0, and decreases back to 0 at x = π/2. The line y = 1/2 intersects this curve exactly at x = -π/3 and x = π/3.</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B