Question

Let $f, g \& h$ be three functions defined as follows :
$f(x)=\frac{32}{4+x^2+x^4}, g(x)=9+x^2 \quad \text { and } \quad h(x)=-x^2-3 x+k$ 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 $h(g(x))<0 \quad \forall x \in R$ is 19 .
• D. Maximum value of $g(f(x))$ is 73 .

Answer 1

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

$f ( x )=\frac{32}{4+ x ^2+ x ^4} ; g ( x )=9+ x ^2 ; h ( x )=- x ^2-3 x + k$
(A) range of $f$ is $(0,8]$
(B) $ h ( f ( x ))>0$ and $h ( g ( x ))<0$
$h (0) \geq 0 \Rightarrow k \geq 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 $g(f(x)) is g(8) = 64 + 9 = 73$.