Question

Find the intervals on which the function f(x)=x^4−4x^3+4x^2 is increasing and/or decreasing

Answer 1

f(x) = x^4 -4x^3 + 4x^2

f ' (x) = 4x^3 - 12x^2 +8x = 4x(x^2 -3x+ 2)

equate f ' (x) to zero, to get critical points

4x(x^2 -3 x+ 2) = 0

x=0,x^2 -3 x+ 2 = 0

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

x = 0,2 and 1

f"(x)=12x^2-24x+8

 f" (0) =  8 > 0 increasing

in the intervals (-∞,0] increasing
f" (1) =  -4 < 0 decreasing

in the intervals [0,1] decreasing
 f" (2) =8 > 0 increasing

in the intervals [2,∞) increasing