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5. Continuity and Differentiation
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Let $f (x)$ and $g (x)$ be two continuous functions defined from $R \rightarrow R$, such that $f (x_1) > f (x_2)$ and $g (x_1) < g (x_2), \forall x_1 > x_2$ , then solution set of $f\,\left( {\,g({\alpha ^2} - 2\alpha )\,} \right) >f\,\left( {\,g(3\alpha - 4)\,} \right)$ is
A
$R$
B
$\phi$
C
$(1, 4)$
D
$R - [1, 4]$
Solution
obviously $f$ is increasing and $g$ is decreasing in $(x_1, x_2)$
hence $f\,\left( {\,g({\alpha ^2} – 2\alpha )\,} \right) >f\,\left( {\,g(3\alpha – 4)\,} \right)$ as $f$ is increasing
==>$g(a^2 – 2\alpha ) > g(3\alpha – 4)$
$\alpha ^2 – 2\alpha < 3\alpha – 4$ as $g$ is decreasing
$\alpha ^2 – 5\alpha + 4 < 0$
$(\alpha – 1)(\alpha – 4) < 0$==>$a \in (1, 4) $
Standard 12
Mathematics
