The range of the function $f(x) = \frac{x+m}{x^2+1}$ $(m \in \mathbb{R})$ contains the interval $[0, 1]$. If $m \geq \frac{3}{k}$, then find $k$.
Step-by-Step Solution
Key Concept: Treating a rational function as an implicit equation in a parameter and requiring non-negative discriminant for real solutions constrains the parameter.
Given $y = \frac{x+m}{x^2+1}$, rearrange to $yx^2 - x + (y-m) = 0$. For $y \in [0,1]$, the discriminant $D = 1 - 4y(y-m) \ge 0$ must hold. This yields $4y^2 - 4my - 1 \le 0$ for all $y \in [0,1]$. Evaluating at endpoints gives $m \ge \frac{3}{4}$ and $k = 4$.
Correct Answer: 4