Trigonometry & Inverse Trigonometry
Inverse Function Inequalities
Grade 12
Question:
<p>The complete set of values of <span class='math'>a</span> for which the function <span class='math'>f(x) = \tan^{-1}(x^2 - 18x + a) \geq 0, \forall x \in \mathbb{R}</span>, is</p>
<p>(a) <span class='math'>(-81, \infty)</span></p>
<p>(b) <span class='math'>[81, \infty)</span></p>
<p>(c) <span class='math'>(-\infty, -81)</span></p>
<p>(d) <span class='math'>(-\infty, -81]</span></p>
Step-by-Step Solution
Key Concept: For tan⁻¹(u) ≥ 0, we need u ≥ 0. The quadratic must be non-negative for all x, which requires non-positive discriminant.
<p>For <span class='math'>\tan^{-1}(x^2 - 18x + a) \geq 0</span> for all <span class='math'>x \in \mathbb{R}</span>, we need <span class='math'>x^2 - 18x + a \geq 0</span> for all <span class='math'>x \in \mathbb{R}</span>.</p><p>The quadratic <span class='math'>x^2 - 18x + a</span> is non-negative for all <span class='math'>x</span> if and only if its discriminant is non-positive: <span class='math'>(-18)^2 - 4(1)(a) \leq 0</span>.</p><p>This gives <span class='math'>324 - 4a \leq 0</span>, so <span class='math'>a \geq 81</span>. Therefore, the answer is <span class='math'>[81, \infty)</span>.</p>
Correct Answer: b