positivity of a function Consider the function $G(\omega):=\omega^4+2r^2\omega^2+b$, where $r,b\in\mathbb{R}$. What constraints on the parameters are needed to guarantee the positivity of $G$ for all $\omega&gt;0$?
First observe that $G(0)&gt;0$ implies $b&gt;0$. On the other hand, if $b&gt;0$ then $G(\omega)\geq b&gt;0$. Hence for $G$ to be positive, it is necessary and sufficient that $b&gt;0$.
First observe that $G(0)&gt;0$ implies $b&gt;0$. On the other hand, if $b&gt;0$ then $G(\omega)\geq b&gt;0$. Hence for $G$ to be positive, it is necessary and sufficient that $b&gt;0$.