A function g is differentiable for all real numbers except at x=2. The graph of g′(x) is described verbally as follows.
• For x<2, g′(x) is a straight line segment with positive slope that crosses the x-axis at x=−1 and is negative to the left of that point and positive to the right of it.
• At x=2, g′(x) has a removable discontinuity (an open circle); the limit of g′(x) as x→2 exists and equals 3.
• For x>2, g′(x) is constant and equal to 3.
(a) Determine the x-coordinates of all critical points of g on the interval −4<x<6. Explain why each point you list is a critical point.
(b) For each critical point found in part (a), decide whether g has a relative maximum, a relative minimum, or neither at that point. Provide a justification based on the sign of g′(x).
(c) On the interval −4<x<6, identify the open intervals where g is concave up and concave down. Explain how your answer is consistent with the behavior of g′(x) described above.