Let g be the rational function given by g(x)=x−1x2+1. The domain of g excludes x=1. Consider g on the interval [−2,4], noting that x=1 creates a break in the domain.
(a) Compute g′(x) and simplify your result.
(b) Identify all x–values in (−2,4) at which g′(x)=0 or is undefined. Combine these with the point of discontinuity x=1 to create the critical set for g on the given interval.
(c) Construct a sign chart for g′(x) on [−2,4], carefully addressing the discontinuity at x=1.
(d) Using the sign chart, determine the intervals on which g is increasing and the intervals on which it is decreasing. Then classify every critical point from part (b) as a local maximum, local minimum, or neither. Provide justification for each classification.
(e) Evaluate g at all critical points from part (b) that lie in the domain as well as at the endpoints x=−2 and x=4. Discuss whether an absolute maximum or absolute minimum of g exists on [−2,4], and support your conclusion.