Question

Let $f,$ $g$ and $h$ be three functions defined as follows :
$f\left(x\right)=\frac{32}{4+x^2+x^4},$ $g\left(x\right)=9+x^2$ and $h\left(x\right)=-x^2-3x+k.$
Which of the following statement(s) is(are) CORRECT?

• A. Maximum value of $g\left(f\left(x\right)\right)$ is $73$.
• B. Number of integers in the range of $f\left(x\right)$ is $8$.
• C. Number of integral values of $k$ for which $h\left(f\left(x\right)\right)>0$ and $h\left(g\left(x\right)\right)<0\forall x\in R$ is $20$.
• D. Number of integral values of $k$ for which $h\left(f\left(x\right)\right)>0$ and $h\left(g\left(x\right)\right)<0\forall x\in R$ is $19$.

Answer 1

Answer: (A), (B), (D)

Number of integral values of $k$ for which $h\left(f\left(x\right)\right)>0$ and $h\left(g\left(x\right)\right)<0\forall x\in R$ is $19$.

$f\left(x\right)=\frac{32}{4+x^2+x^4}$
Clearly, domain of $f$ is $R.$
Since $x^4+x^2+4\in [4,\infty ),$
$\therefore$ Range of $f$ is $(0,8]$

$h\left(f\left(x\right)\right)>0$ and $h\left(g\left(x\right)\right)<0$
Since $f\left(x\right)\in (0,8]$ and $g\left(x\right)\in [9,\infty ),$
$h\left(0\right)\ge 0\Rightarrow k\ge 0$
$h\left(8\right)>0\Rightarrow -64-24+k>0\Rightarrow k>88$
$h\left(9\right)<0\Rightarrow -81-27+k<0\Rightarrow k<108$
Hence, $k\in \{89,90,\dots ,106,107\}$
Number of integral values of $k$ is $19$.

Since $f\left(x\right)\in (0,8],$
$\therefore$ Maximum value of $g\left(f\left(x\right)\right)$ is $g\left(8\right)=64+9=73$