Question

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

• A. No real value of b &c
• B. $0<c<b\sqrt{2}$
• C. $|c|<|b|\sqrt{2}$
• D. $|c|>|b|\sqrt{2}$

Answer 1

The correct option is D

$|c|>|b|\sqrt{2}$

We have
$f\left(x\right)=x^2+2bx+2c^2;g\left(x\right)=-x^2-2cx+b^2\Rightarrow f\left(x\right)=(x+b{)}^2+2c^2-b^2$
and $g\left(x\right)=-(x+c{)}^2+b^2+c^2$
$\Rightarrow f_{min}=2c^2-b^2$ and $g_{max}=b^2+c^2$
For $f_{min}>g_{max}\Rightarrow 2c^2-b^2>b^2+c^2$
$\Rightarrow c^2>2b^2\Rightarrow |c|>|b|\sqrt{2}$