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Q.
Number of solution(s) satisfying the equation, $3 x^2-2 x^3=\log _2\left(x^2+1\right)-\log _2 x$ is -
Application of Derivatives
Solution:
$3 x ^2-2 x ^3=\log _2\left( x +\frac{1}{ x }\right), x >0 $
$ f ( x )=3 x ^2-2 x ^3 $
$ f ^{\prime}( x )=6 x -6 x ^2 $
$=6 x (1- x )$
$f ( x ) \leq f (1) $
$ f ( x ) \leq 1 $
$\Rightarrow \text { LHS } \leq 1 \text { \& RHS } \geq 1$
$\text { LHS }=\text { RHS }=1 \text { for } x =1$
$\therefore$ equation has exactly one solution
