A differentiable function g is defined for −4<x<6. Its first derivative satisfies g′(1)=0 and g′(x)=0 for all x=1 in the interval. The second derivative is known to be positive on (−4,1) and negative on (1,6). No other information about g is given.
(a) Describe the concavity of the graph of g on the intervals (−4,1) and (1,6).
(b) Determine whether x=1 is a local maximum or a local minimum of g. Justify your answer.
(c) Explain whether x=1 must also be an absolute extremum of g on (−4,6). Provide a complete justification based on the given information.