MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
Let r(x)=x5−5x3+4x for x in the closed interval [−3,3].
(1) Show that r is continuous on [−3,3]. Provide a concise but complete justification.
(2) Compute r(−3) and r(3), and determine which is the minimum and which is the maximum value of r on [−3,3]. Explain your reasoning fully, including any derivative tests you employ.
(3) Prove that for every real number d satisfying r(−3)≤d≤r(3) there exists at least one number c in [−3,3] such that r(c)=d. Your proof should be written as a logically coherent argument using the properties established in parts (a) and (b).
(4) Analyze whether the equation r(x)=0 must have more than one solution in [−3,3]. Support your conclusion with detailed justification, referencing the structure of r and its derivative as needed.