SOLVED:The three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions f , g , and h whos values at 0 are all 0 and, for x ≠ 0 , f(x) = x^4 sin(1)/(x) g(x) = x^4 (2 + sin(1)/(x) ) h(x) = x^4 (-2 + sin(1)/(x) ) (a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0 . (b) Show that f has neither a local maximum nor a local minimum at 0 , g has a local minimum, and h has a local maximum.

Harvey Mudd College

Baylor University

University of Nottingham

Idaho State University

This textbook answer is only visible when subscribed! Please subscribe to view the answer