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\, ∀ \, x \in R. $If g is a function defined on R with values in the interval (0, ∞) such that $f(x)= ln $ {g(x)} $∀ \, x \in R, $ then the point in R at which f(x) has a local maximum, is

2009

The correct answer is option (3) : 2009

We have,

$f(x) = ln$ {g(x)}

$⇒g(x) =e^{f(x)}$

$⇒g'(x) =e^{f(x)}f'(x)$

At points of local maximum of g(x), we must have

$g'(x) = 0 $

$⇒e^{f(x)}f'(x) = 0 $

$⇒f'(x) = 0 $

$⇒2010 (x-2009) (x- 2010)^2 (x-2011)^3(x-2012)^4=0$

$⇒x= 2009 , 2010, 2011, 2012 $

The changes in signs of f'(x) for different values of x are as shown above.

So, f(x) attains a local maximum at x = 2009 only.