Question

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

• A. no relation
• B. 0 < c <
• C. |c| <
• D. |c| < |b|

Answer 1

The correct option is D

|c| < |b|

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

As min f(x) > max g (x), we get $2c^2-b^2$ > $b^2+c^2$

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