A twice–differentiable function f is defined for −4≤x≤6. The following verbal description summarizes key features of the graph of f.
• The function decreases on [−4,−1], attaining a relative minimum of 2 at x=−1.
• It increases on [−1,2], attaining a relative maximum of 5 at x=2.
• It then decreases on [2,6].
• The slopes of the tangent lines change continuously, and the steepest positive tangent occurs at x=1.
• The point of inflection closest to the origin is at x=0.
Based on this description, answer the following questions about f′, the first derivative of f.
(a) Describe the sign of f′(x) on each of the intervals (−4,−1),(−1,2),(2,6). Give a clear verbal explanation in complete sentences.
(b) List all x–values in the interval −4≤x≤6 at which f′(x)=0. For each value, state whether it corresponds to a relative maximum or minimum of f and justify your answer.
(c) Explain how the location of the point of inflection of f at x=0 influences the behavior of f′ near x=0. Your answer should reference how the graph of f′ changes from the left side of x=0 to the right side.