Let p be the quartic function p(x)=x4−4x2+1 on the closed interval [−2,2].
a. Explain why the hypotheses guaranteeing the existence of at least one real number c in (−2,2) with p′(c)=2−(−2)p(2)−p(−2) are satisfied. Cite each hypothesis explicitly.
b. Calculate the average rate of change of p on [−2,2]. Express your answer exactly.
c. Solve the equation p′(c)=2−(−2)p(2)−p(−2). Provide all real solutions for c in (−2,2), accurate to three decimal places. Show your algebraic steps and clearly indicate any numerical methods used in reaching your approximations.
d. Confirm that every solution obtained in part (c) indeed lies within (−2,2) and discuss why no other solutions can occur in that interval.