Question

If $1$ lies between the roots of the equation $y^2-my+1=0$ and $\left[x\right]$ is the GIF function,
then the value of $\left[{\left(\frac{4|x|}{|x{|}^2+16}\right)}^m\right]$, is

• A. $0$
• B. $1$
• C. $2$
• D. None of the above.

Answer 1

The correct option is A $0$

Given that $1$lies between roots of $y^2-my+1=0=f\left(y\right)$

As co efficient of $y^2=1>0$

$\therefore f\left(1\right)<0$

$1-m+1<0$

$\therefore m>2$

${\left|x\right|}^2-4\left|x\right|+16=g\left(x\right)$

If $x<0$

$g\left(x\right)=x^2+4x+16=0$

$△=16-64=-48<0$

$\therefore g\left(x\right)>0\forall x<0$

If $x\ge 0$

$g\left(x\right)=x^2-4x+16=0$

$△=16-64=-48<0$

$\therefore g\left(x\right)>0\forall x\ge 0$

$\therefore g\left(x\right)={\left|x\right|}^2-4\left|x\right|+16>0\forall xϵR$

$\therefore {\left|x\right|}^2+16>4\left|x\right|$

$0<\frac{4\left|x\right|}{{\left|x\right|}^2+16}<1$

$0<{\left(\frac{4\left|x\right|}{{\left|x\right|}^2+16}\right)}^m=0,\left[x\right]$ is GIF function.