Question

Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f(x)=2010(x-2009)(x-2010)^{2}$ $(x-2011)^{3}(x-2012)^{4}$, for all $x \in R$. If $g$ is a function defined on $R$ with values in the interval $(0, \infty)$ such that $f(x)=\ln (g(x))$, for all $x \in R$, then the number of points in $R$ at which $g$ has a local maximum is

Answer 1

$f ( x )=\ln \{ g ( x )\}$
$g ( x )= e ^{ f (x)}$
$g '( x )= e ^{ f ( s )} \cdot f '( x )$
$g '( x )=0 $
$\Rightarrow f '( x )=0$ as $e ^{ f ( s )} \neq 0$
$\Rightarrow 2010( x -2009)( x -2010)^{2}( x -2011)^{3}( x -2012)^{4}=0$
so there is only one point of local maxima.