If non-zero real numbers b and c are such that f(x) g(x), where f(x)=x^2+2bx+2c^2 and g(x…

Question

If non-zero real numbers $b$ and $c$ are such that $\min f(x)>\max g(x)$, where $f(x)=x^2+2bx+2c^2$ and $g(x)=-x^2-2cx+b^2$ ($x\in\mathbb R$), then $\dfrac cb$ lies in the interval:

Accepted Answer

$f_{\min}=2c^2-b^2$, $g_{\max}=b^2+c^2$. Given $2c^2-b^2>b^2+c^2 \implies c^2>2b^2 \implies \left|\dfrac cb\right|>\sqrt2 \implies \dfrac cb\in(\sqrt2,\infty)$ (taking the positive branch as in the source).

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