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

$f\left(x\right)={\left(x+b\right)}^2+2c^2-b^2,$

$g\left(x\right)=b^2+c^2-{\left(x+c\right)}^2$

$\Rightarrow$      min. $f\left(x\right)=2c^2-b^2,$    max. $g\left(x\right)=b^2+c^2$

$\Rightarrow$      $2c^2-b^2>b^2+c^2\Rightarrow c^2>2b^2$

$\Rightarrow$       $\left|c\right|>\left|b\right|\sqrt{2}$