Let be the quartic function on the closed interval .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let pp be the quartic function p(x)=x4−4x2+1p(x)=x^{4}-4x^{2}+1 on the closed interval [−2,2][-2,2]. (1) Explain why the hypotheses guaranteeing the existence of at least one real number cc in (−2,2)(-2,2) with p′(c)=p(2)−p(−2)2−(−2)p'(c)=\dfrac{p(2)-p(-2)}{2-(-2)} are satisfied. Cite each hypothesis explicitly. (2) Calculate the average rate of change of pp on [−2,2][-2,2]. Express your answer exactly. (3) Solve the equation p′(c)=p(2)−p(−2)2−(−2)p'(c)=\dfrac{p(2)-p(-2)}{2-(-2)}. Provide all real solutions for cc in (−2,2)(-2,2), accurate to three decimal places. Show your algebraic steps and clearly indicate any numerical methods used in reaching your approximations. (4) Confirm that every solution obtained in part (c) indeed lies within (−2,2)(-2,2) and discuss why no other solutions can occur in that interval.

Related theory

Browse the problem library