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

Engineering

Mathematics

Maxima and Minima

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

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g (x) = e^{f (x)}

g ' (cx) = e^{f (x)} f ' (x)

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

g ' (x) = 0 $\Rightarrow$ x = 2009, 2010, 2011, 2012

for max. g ' (x) must change sign from + ve to – ve

x = 2009

$∴ 1$

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