Question

Let $f(x)$ and $g(x)$ are two continuous functions from $R\to R$ such that $f\left(x_1\right)>f\left(x_2\right)$ and $g\left(x_1\right)<g\left(x_2\right)$ for all $x_1>x_2$ 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)
$\Rightarrow {\alpha }^2-5\alpha +4<0$
$\Rightarrow \left(\alpha -1\right)\left(\alpha -4\right)<0$
$\Rightarrow \alpha \in \left(1,4\right)$