Question

Given the function g(x) = b(−5x + 1)6 − a , where a ≠ 0 and b ≠ 0 are constants.

A. Find g′(x) and g′′(x).

B. Prove that g is monotonic (this means that either g always increases or g always decreases).

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

Answer 1

a.
g'(x) = 6b(-5x + 1)^5 (-5)
g'(x) = -30b(-5x +1)^5

g''(x) = -30b(5)(-5x + 1)^4 (-5)
g''(x) = 750b (-5x +1)^4

b.
g(x) = b(−5x + 1)6 − a
when
g(-x) = b(5x +1)6 - a

c.
g'(x) = -30b(-5x +1)^5 = 0
-5x +1 = 0
x = 15