Question

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

Answer 1

$f^{\prime }\left(x\right)=2010(x-2009)(x-2010{)}^2(x-2011{)}^2(x-2012{)}^4$

$f^{\prime }\left(x\right)=0$

$2010(x-2009)(x-2010{)}^2(x-2011{)}^2(x-2012{)}^4=0$

$x=2009,2010,2011,x=2012$

Clearly it has $4$ points of net value.