Find κ and λ $\in R$ for which $\frac {x^2+κx+λ}{x^2+1} \leq 2$ for all $x \in R$.

Question

By doing the math, we get

$x^2-κx+2-λ \geq 0$.

Also $D = κ^2 - 4(2-λ)$

...but I don't know how to continue!

Answer 1

hint: $D \le 0$ is all you need !

Answer 2

Now, $$\kappa^2-4(2-\lambda)\leq0$$ or $$\kappa^2+4\lambda\leq8.$$

Answer 3

$x^2+1 > 0$ so you can multiply the equation by the denominator. Then you get a new quadratic equation $q(x)\geq 0$, i.e. a parabola above the x-axis. Investigate when this is the case.