If  $f\left(x\right)=x^2+2bx+2c^2$   and  $g\left(x\right)=-x^2-2cx+b^2$  are such that min f(x)>max g(x), then relation between b and c, is:

 $f\left(x\right)={(x+b)}^2+2c^2-b^2\Rightarrow \min f\left(x\right)=2c^2-b^2$
Also  $g\left(x\right)=-x^2-2cx+b^2=b^2+c^2-{(x+c)}^2\Rightarrow \max g\left(x\right)=b^2+c^2$
As  $\min f\left(x\right)>\max g\left(x\right)$  we get  $2c^2-b^2>b^2+c^2$
 $\Rightarrow c^2>2b^2\Rightarrow \left|c\right|>\sqrt{2}\left|b\right|$