Let f(x) and g(x) be two continuous functions defined from such that f (x_1 ) > f (x_2 ) and , then solution set of is.

JEE MainMathematicsToughMonotonocity

Let f(x) and g(x) be two continuous functions defined
from $\ R \rightarrow R,$ such that
f (x_1 ) > f (x_2 ) and
$g\left( x_{1} \right) < g\left( x_{2} \right),\forall{\ x}_{1} > x_{2}$ , then
solution set of
$f\left( g\left( \alpha^{2} - 2\alpha \right) \right) > \ f(g(3\alpha - 4))$
is.

Show answer and solution

Obviously f is increasing and g is decreasing in (x_1, x_2)
Hence,

f\left( g\left( \alpha^{2} - 2\alpha \right) \right) > f(g(3\alpha - 4))

as f is
increasing

\Rightarrow

g\left( a^{2} - 2\alpha \right) > g(3\alpha - 4)

\therefore

\alpha^{2} - 2\alpha < 3\alpha - 4

as g is decreasing

\alpha^{2} - 5\alpha + 4 < 0

(\alpha - 1)(\alpha - 4) < 0

\Rightarrow \alpha \in (1,4)\rbrack

Source: Grass-G / M-12 Mathematics Paper 2 Nov.docx

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