Consider the function g defined on the closed interval [−1,1] by g(x)=e2x+x.
a. Justify that g meets all hypotheses required to guarantee at least one point c in (−1,1) satisfying g′(c)=1−(−1)g(1)−g(−1). Clearly state each hypothesis and how it is satisfied.
b. Compute the average rate of change of g on [−1,1] and express your answer exactly.
c. Solve g′(c)=1−(−1)g(1)−g(−1) for all c in (−1,1). Give exact values when possible or justify any numerical approximation used.
d. Identify which of the c-values obtained in part (c) also satisfy the additional condition c>0, and state your final answer(s).