Question

Let $f(x)$ and $g(x)$ are two continuous functions from $R \rightarrow R$ such that $f\left(x_{1}\right)>f\left(x_{2}\right)$ and $g\left(x_{1}\right)x_{2}$ and if $f\left(g\left(\alpha^{2}-2 \alpha\right)\right)>f(g(3 \alpha-4))$, then the complete set of values of $\alpha$ is

• A. $\left(1 , 4\right)$
• B. $\left(4 , 10\right)$
• C. $\left(- 4 , 10\right)$
• D. $\left(- 1 , 4\right)$

Answer 1

Answer: (A)

Here $f\left(g\left(\alpha^{2}-2 \alpha\right)\right)>f(g(3 \alpha-4))$
$ \Rightarrow g\left(\alpha^{2}-2 \alpha\right)>g(3 \alpha-4)(\because f \text { is increasing }) $
$\Rightarrow \alpha ^{2}-2\alpha < 3\alpha -4$ ( $\because $ $g$ is decreasing)
$\Rightarrow \alpha ^{2}-5\alpha +4 < 0$
$\Rightarrow \left(\alpha - 1\right)\left(\alpha - 4\right) < 0$
$\Rightarrow \alpha \in \left(1 , 4\right)$