<p>Find the complete set of values of <span style="font-style: italic;">a</span> for which <span style="font-style: italic;">f</span>(θ) > 0 for all θ ∈ [0, π/2), where <span style="font-style: italic;">f</span>(θ) = tan²θ + (<span style="font-style: italic;">a</span> + 1)tanθ – (<span style="font-style: italic;">a</span> – 3)</p>
Step-by-Step Solution
Key Concept: Substitute t = tanθ and analyze the quadratic inequality on the interval [0, ∞) by checking critical conditions at the boundary and vertex.
<p><strong>Solution:</strong> Let <span style="font-style: italic;">t</span> = tanθ. Since θ ∈ [0, π/2), we have <span style="font-style: italic;">t</span> ∈ [0, ∞). The function becomes <span style="font-style: italic;">g</span>(<span style="font-style: italic;">t</span>) = <span style="font-style: italic;">t</span>² + (<span style="font-style: italic;">a</span> + 1)<span style="font-style: italic;">t</span> – (<span style="font-style: italic;">a</span> – 3).</p><p>For <span style="font-style: italic;">f</span>(θ) > 0 ∀ θ ∈ [0, π/2), we need <span style="font-style: italic;">g</span>(<span style="font-style: italic;">t</span>) > 0 ∀ <span style="font-style: italic;">t</span> ∈ [0, ∞).</p><p>At <span style="font-style: italic;">t</span> = 0: <span style="font-style: italic;">g</span>(0) = –(<span style="font-style: italic;">a</span> – 3) = 3 – <span style="font-style: italic;">a</span>. For <span style="font-style: italic;">g</span>(0) > 0, we need <span style="font-style: italic;">a</span> < 3.</p><p>The vertex is at <span style="font-style: italic;">t</span> = –(a+1)/2. If this is non-negative, we need the minimum value to be positive. Otherwise, we need the roots to be negative.</p><p>∴ The answer is (B): (–∞, –3)</p>
Correct Answer: B