Question

Let $f$ be a function defined on $R$ such that
$f\text{'}\left(x\right)=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 $f\left(x\right)=\ln \left(g\right(x\left)\right)$ for all $x\in R$, then the number of points in $R$ at which $g$ has a local maximum is

Answer 1

$f\left(x\right)=\ln \left\{g\right(x\left)\right\}$
$\therefore g\left(x\right)=e^{f\left(x\right)}$
$\therefore g^{\prime }\left(x\right)=e^{f\left(x\right)}.f\left(x\right)$
$g^{\prime }\left(x\right)=0\Rightarrow f^{\prime }\left(x\right)=0$ as $e^{f\left(x\right)}\ne 0$
or $2010(x-2009)(x-2010{)}^2(x-2011{)}^3(x-2012{)}^4=0$
So there is only one point of local maxima