12. The graph of \( f' \), the derivative of \( f \), is shown in the figure above. Which of the following describes all relative extrema of \( f \) on the open interval \( (a, b) \)?






Answer is: option2

Two relative maxima and one relative minimum

Solution:

Critical points occur where \( f'(x) = 0 \), i.e., where the curve crosses the x-axis.
From the graph, there are four points where \( f'(x) \) crosses the x-axis.

To classify these critical points, we analyze the sign changes of \( f'(x) \):

  1. First Crossing: \( f'(x) \) changes from positive to negative → Relative Maximum
  2. Second Crossing: \( f'(x) \) changes from negative to positive → Relative Minimum
  3. Third Crossing: \( f'(x) \) remains positive before and after → Not an extremum
  4. Fourth Crossing: \( f'(x) \) changes from positive to negative → Relative Maximum

Two relative maxima

One relative minimum

The correct answer is - Two relative maxima and one relative minimum.

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