Final answer:
Without sufficient detail on the function f(x), we cannot definitively determine the local maximum. However, given that at x=3 the function has a positive and decreasing slope, x=3 is the most likely candidate for a local maximum.
Step-by-step explanation:
To determine where the function f has a local maximum, we analyze the given options and information provided. At x=3, the function f(x) has a positive slope that is decreasing, suggesting that the slope may become zero or negative past this point, which could indicate a local maximum. Option 1 (x=1) and option 2 (x=2) are less likely to be maxima without additional information on the slope or function values at those points.
From the data, we unfortunately cannot conclude with certainty where the local maximum is, as we would require specific function details, derivatives, or graphical analysis to identify it. However, if we use the given clue that at x=3 the function has a positive and decreasing slope, this might suggest that x=3 is the point right before the slope reaches zero, making it the most likely candidate for a local maximum of the options provided. Without further information or the actual loss of slope beyond x=3, we can't definitively assign option 3 (x=3) as the local maximum, yet it is logically the most probable of the given choices.