Question

Let $f,g\&h$ be three functions defined as follows :
$f(x)=\frac{32}{4+x^2+x^4},g(x)=9+x^2\text{ and }h(x)=-x^2-3x+k$ Identify which of the following statement(s) is(are) correct?

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

Answer 1

Answer: (A), (C), (D)

$f(x)=\frac{32}{4+x^2+x^4};g(x)=9+x^2;h(x)=-x^2-3x+k$
(A) range of $f$ is $(0,8]$
(B) $h(f(x))>0$ and $h(g(x))<0$
$h(0)\ge 0\Rightarrow k\ge 0$
$h(8)>0\Rightarrow -64-24+k>0\Rightarrow k>88$
$h(9)<0\Rightarrow -81-27+k<0\Rightarrow k<108$
Number of integral values of $k$ is 19 .

(D) Maximum value of $g(f(x))isg(8)=64+9=73$.