Question

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 for all x in R. If g is a function on R with values in the interval (0, ∞) such that f(x) = ln (g(x)) for all x in R then the number of points in R at which g has a local minimum is...

Answer 1

Given f(x) = ln(g(x))

⇒ g(x) = e^f(x)

⇒ g'(x) = e^f(x) .f'(x)

For local max or min, g'(x) = 0 gives

⇒ e^f(x) .f'(x) = 0

⇒ f'(x) = 0

⇒  f'(x) = 2010(x – 2009)(x – 2010)² . (x – 2011)³ (x – 2012)⁴ = 0

⇒ x = 2009, 2010, 2011, 2012

From, the sign scheme, it is clear that,

f(x) has a local maximum at x = 2009.

and local minimum at x = 2011.