HELP please with this question!!?

Question

Let f(x) = (1 - 2x)^2/(1 - 4x)^3. Then f ‘(x) = 8(1 - 2x)(1- x)/(1 - 4x)^4. When is f(x) increasing?

a) x < ½, x > 1
b) x < ½
c) ½ < x < 1
d) x < ¼, ¼ < x < ½, x > 1

Answer 1

f(x) is increasing when f '(x) is positive.

Since f '(x) is zero at 1/2 and 1, check values before 1/2, between 1/2 and 1 and after 1.

f '(0) = 8
f '(3/4) = -1/16

f ' (3/2) = 8/625

This function is increasing when x < 1/2 (on both sides of the vertical asymptote x = 1/4... with a discontinuity at 1/4) and when x > 1.

answer choice d) is correct

Answer 2

denominator always positive so search only signe of numerator

it is positive for 1/2

Answer 3

(1 - 2x)(1- x)>0

2x^2-3x+1>0

a) x < ½, x > 1