Given equation f(x) = x^4 + 4x^3 - 2 so f ' (x) = 4x^3 + 12x^2 which simplified to f ' (x) = 4x^2 (x+3), thus our critical points are 0 and -3 I will use this to sketch the curve f ' (x) = 4x^2 (x+3) Results are below. According to my "sketch", the sequence is - , +, +, + B means "before" A means "after" I used a very close number to generate this list But I could not generate any sketch because (0,0) pair cannot have a +, +, + sequence. There must be a - sign, meaning the correct sequence should be -, +, -, +. I need to hite (0,0) but there is no way as (-3,0) is increasing. My calculator shows a different sequence. What is wrong with my sketch logic here?