Mathematics - Functions Question with Solution | TestHub

We have, $f(x)=x^2+2bx+2c^2;g(x)=-x^2-2cx+b^2$ $\Rightarrow \mathrm{f}(\mathrm{x})=(\mathrm{x}+\mathrm{b}{)}^2+2{\mathrm{c}}^2-{\mathrm{b}}^2$ and, $g(x)=-(x+c{)}^2+b^2+c^2$ $\Rightarrow {\mathrm{f}}_{\text{min }}=2{\mathrm{c}}^2-{\mathrm{b}}^2$ and ${\mathrm{g}}_{\text{max }}={\mathrm{b}}^2+{\mathrm{c}}^2$ for, ${\mathrm{f}}_{\text{min }}>{\mathrm{g}}_{\text{max }}\Rightarrow 2{\mathrm{c}}^2-{\mathrm{b}}^2>{\mathrm{b}}^2+{\mathrm{c}}^2$ $\Rightarrow c^2>2b^2\Rightarrow |c|>|b|\sqrt{2}$