Question

Consider the function h(x) = a(-2x+1)^5-b, where a doesn't equal 0 and b doesn't equal 0 are constants.

A. Find h'(x) and h"(x)

B. show that h is monotonic ( that is, that either h always increases or remains constant or h always decreases or remains constant)

C. Show that x-coordinate(s) of the location(s) of the critical points are independent of a and b.

Answer 1

h' = 5a(-2x+1)^4 (-2) = -10a(-2x+1)^4
h'' = 80a(-2x+1)^3

Since h' >=0 for all x, it is monotonic

h'=0 ==> (-2x+1)^4 = 0, independent of a,b