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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 ∈ R. If g is a function defined on R with values in the interval (0, ∞) such that f(x) = ln(g(x)), for all x ∈ R, then the number of points in R at which g has a local maximum is 0, 1, 2, or 3?

1

0

2

3

Solution:

f(x) = ln(g(x))
g(x) = e^(f(x))
g'(x) = e^(f(x)) * f'(x)
g'(x) = 0 ⇒ f'(x) = 0 since e^(f(x)) ≠ 0
⇒ 2010(x - 2009)(x - 2010)²(x - 2011)³(x - 2012)⁴ = 0
So there is only one point of local maxima.