Question

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

• A. No relation
• B. $0<c<\frac{b}{2}$
• C. $|c|<\sqrt{2}|b|$
• D. $|c|>\sqrt{2}|b|$

Answer 1

Answer: (D)

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

We have, $f\left(x\right)=x^2+2bx+2c^2$
$\Rightarrow \text{min}f\left(x\right)=2c^2-b^2$
and $g\left(x\right)=b^2+c^2-(x+c{)}^2$
$\Rightarrow \text{max}g\left(x\right)=b^2+c^2$
Thus, $\text{min}f\left(x\right)>\text{max}g\left(x\right)$
$\Rightarrow 2c^2-b^2>b^2+c^2$
$\Rightarrow c^2>2b^2$

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