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