A differentiable function g is defined for all real numbers. The sign of its first derivative g′(x) is summarized below:
Interval : (−∞,−1), −1, (−1,2), 2, (2,∞)
Sign of g′(x): +, 0, −, 0, +
No other critical numbers exist.
(a) Determine the intervals on which g is increasing and the intervals on which g is decreasing. Give a brief justification based on the information provided.
(b) Identify each x–value at which g attains a local extremum and classify each extremum as a local maximum or a local minimum. Support your answer with appropriate reasoning.
(c) Explain why the information in the table is sufficient to guarantee the classifications you provided in part (b), even though no explicit formula for g(x) has been supplied.