Let f be a function defined on R such that
`f'(x) = 2010(x - 2009) (x - 2010)^2 (x - 2011)^3 ( x - 2012)^4 AA x in R`
If g is a function defined on R with values in the interval ] `0,oo`[ such that `f(x) = log (g (x)) AA x in R` then the number of point in R at which g has a local maximum is

Options

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

Answer

To solve the problem, we need to analyze the function \( f'(x) \) given by:

\[ f'(x) = 2010(x - 2009)(x - 2010)^2(x - 2011)^3(x - 2012)^4 \]

### Step 1: Find the critical points of \( f(x) \)

To find the critical points where \( g(x) \) has a local maximum, we need to set \( f'(x) = 0 \):

\[ 2010(x - 2009)(x - 2010)^2(x - 2011)^3(x - 2012)^4 = 0 \]

The critical points occur when any of the factors equal zero:

1. \( x - 2009 = 0 \) → \( x = 2009 \)
2. \( x - 2010 = 0 \) → \( x = 2010 \)
3. \( x - 2011 = 0 \) → \( x = 2011 \)
4. \( x - 2012 = 0 \) → \( x = 2012 \)

### Step 2: Determine the multiplicity of each critical point

The multiplicity of each root is important because it affects whether \( g(x) \) has a local maximum or minimum at those points:

- \( x = 2009 \): multiplicity 1 (simple root)
- \( x = 2010 \): multiplicity 2 (double root)
- \( x = 2011 \): multiplicity 3 (triple root)
- \( x = 2012 \): multiplicity 4 (quadruple root)

### Step 3: Analyze the behavior of \( f'(x) \) around the critical points

To determine whether each critical point corresponds to a local maximum or minimum for \( g(x) \), we analyze the sign of \( f'(x) \) around these points:

- For \( x < 2009 \): \( f'(x) > 0 \) (increasing)
- For \( 2009 < x < 2010 \): \( f'(x) < 0 \) (decreasing)
- For \( 2010 < x < 2011 \): \( f'(x) > 0 \) (increasing)
- For \( 2011 < x < 2012 \): \( f'(x) < 0 \) (decreasing)
- For \( x > 2012 \): \( f'(x) > 0 \) (increasing)

### Step 4: Identify local maxima and minima

From the analysis:

- At \( x = 2009 \): \( f(x) \) changes from increasing to decreasing (local maximum).
- At \( x = 2010 \): \( f(x) \) does not change direction (local minimum).
- At \( x = 2011 \): \( f(x) \) changes from increasing to decreasing (local maximum).
- At \( x = 2012 \): \( f(x) \) does not change direction (local minimum).

### Conclusion

Thus, the points where \( g(x) \) has a local maximum are at \( x = 2009 \) and \( x = 2011 \). Therefore, the number of points in \( \mathbb{R} \) at which \( g \) has a local maximum is:

\[ \boxed{2} \]