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. $0>c>\frac{b}{2}$
• C. $|c|>2|b|$
• D. $|c|>\sqrt{2}|b|$

Answer 1

The correct option is D $|c|>\sqrt{2}|b|$

$f\left(x\right)=x^2+2bx+2c^2$

$=(x+b{)}^2+2c^2-b^2$

$\therefore minf\left(x\right)=2c^2-b^2$

$g\left(x\right)=-x^2-2cx+b^2$

$=-(x-c{)}^2+c^2+b^2$

$\therefore maxf\left(x\right)=c^2+b^2$

$minf\left(x\right)>maxg\left(x\right)$

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

$\therefore c^2>2b^2$

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