Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>If the range of the function \(f(x) = \tan^{-1}(3x^2 + bx + c)\) is \(\left[0, \frac{\pi}{2}\right)\) (domain is \(\mathbb{R}\)), then:</p>
<p>(a) \(b^2 = 3c\)</p>
<p>(b) \(b^2 = 4c\)</p>
<p>(c) \(b^2 = 12c\)</p>
<p>(d) \(b^2 = 8c\)</p>
Step-by-Step Solution
Key Concept: For the range of \(\tan^{-1}\) to be \([0, \pi/2)\), the argument \(3x^2 + bx + c\) must have minimum value 0. Use the vertex form of quadratic.
<p>Solution not provided in source text.</p>
Correct Answer: c