Question

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

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

Answer 1

The correct option is C

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

$f\left(x\right)=(x+b{)}^2+2c^2-b^2\Rightarrow minf(x)=2c^2-b^{2}Also,g(x)=-x^2-2cx+b^2=b^2+c^2-(x+c{)}^2\Rightarrow maxg\left(x\right)=b^2+c^2$

As $minf\left(x\right)>maxg\left(x\right)$, we get:
$2c^2-b^2>b^2+c^2$
$\Rightarrow c^2>2b^2$
$\Rightarrow |c|>\sqrt{2}|b|$