Let `f` be a function defined on `R` (the set of all real numbers) such that `f^(prime)(x)=2010(x-2009)(x-2010)^2(x-2011)^3(x-2012)^4,` for all `x in Rdot` If `g` is a function defined on `R` with values in the interval `(0,oo)` such that `f(x)=ln(g(x)),` for all `x in R ,` then the number of point is `R` at which `g` has a local maximum is ___

Options

A. 1
B. 2
C. 3
D. 4

Answer

A

Explanation

We have,
f(x)=In {g(x)}
` g(x) =e^(f(x))`
`g(x)=e^(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`
`rArr 2010 (x-2009)(x-2010)^2(x-2011)^3(x-2012)^4 =0`

`rArr x= 2009,2010,2011,2012 `
The change 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