Question

Let $f(x)$ and $g(x)$ be two continuous functions defined from $R\to R$, such that $f\left(x_1\right)>f\left(x_2\right)$ and $g\left(x_1\right)>g\left(x_2\right)$ $\forall x_1>x_2$. Then the solution set of $f\left(g\left({\alpha }^2-2\alpha \right)\right)>f(g(3\alpha -4))$, is

• A. $\alpha \in [1,4)$
• B. $\alpha \in (1,4)$
• C. $\alpha \in (1,4]$
• D. None of these

Answer 1

Answer: (B)

Obviosuly, $f$ is increasing and $g$ is decreasing in $R$.
Hence, $f\left(g\left({\alpha }^2-2\alpha \right)\right)>f(g(3\alpha -4))$
or $g\left(a^2-2\alpha \right)>g(3\alpha -4)$
$(\because f$ is increasing $)$
or ${\alpha }^2-2\alpha <3\alpha -4$
(As $g$ is decreasing)
or ${\alpha }^2-5\alpha +4<0$ or
$(\alpha -1)(\alpha -4)<0$ or $\alpha \in (1,4)$