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

$f\,'(x)=2010\,(x-2009)(x-2010)^{2}(x-2011)^{3}(x-2012)^{4}$

$f(x)=\ln(g(x)),\qquad g(x)>0$

$\text{Find: number of points where } g \text{ has a local maximum}$

$f=\ln g \Rightarrow \dfrac{g'}{g}=f\,' \Rightarrow g'=g\,f\,'$

$g>0 \Rightarrow \operatorname{sign}(g')=\operatorname{sign}(f\,')$

$f\,'=2010\,(x-2009)(x-2010)^{2}(x-2011)^{3}(x-2012)^{4}$

$(x-2010)^{2}\ge0,\ (x-2012)^{4}\ge0 \Rightarrow \text{no sign change}$

$\operatorname{sign}(f\,')=\operatorname{sign}\big((x-2009)(x-2011)^{3}\big)$

$f\,'(x)=0 \Rightarrow x=2009,\ 2010,\ 2011,\ 2012$

$x<2009:\ f\,'>0$

$2009<x<2011:\ f\,'<0$

$x>2011:\ f\,'>0$

$x=2009:\ +\to-\ \Rightarrow g \text{ has local max}$

$x=2011:\ -\to+\ \Rightarrow g \text{ has local min}$

$x=2010,\ 2012:\ \text{no sign change} \Rightarrow \text{neither}$

$\text{Number of local maxima of } g = 1$