Let f be the function defined by f(x)=xe−∣x∣+1 for −2≤x≤4.
Answer the questions below to analyze the extrema of f on the given interval.
(a) Verify that f satisfies all hypotheses of any theorem that would guarantee the existence of both an absolute maximum and an absolute minimum on [−2,4]. Provide a brief justification.
(b) Compute f′(x) for −2<x<0 and for 0<x<4. Show all work that leads to your derivative expressions.
(c) Determine all critical points of f in the interval (−2,4). Include points where f′(x)=0 and points where f′(x) does not exist.
(d) Using the values of f at the endpoints and at the critical points, identify the absolute maximum and absolute minimum values of f on [−2,4] and the x-value(s) where they occur. Support your conclusions with clear reasoning.