關於Differentiation的簡單A.Maths.問題

For each of the following functions:
(a) determine the range of values of x that the function is
(i) increasing,
(ii) decreasing;
(b) find the maximum and minimum points.

1. y = x^6 – 3x^2

dy / dx = 6x^5 – 6x
= 6x(x^2 + 1)(x + 1)(x – 1)

when dy / dx = 0,
6x(x^2 + 1)(x + 1)(x – 1) = 0
x = 0, 1 or -1





X


X < -1


X = -1


-1 <x<0


X = 0


0 < x <1


X =1


X > 1



Dy / dx


-


0


+


0


-


0


+


(a) (i) increasing: [-1, 0], [1, ∞]←此方括是為圓括。
(ii) decreasing: (-∞, -1) ←此圓括是為方括。, [0, 1]

(b) maximum point: (0, 0)
minimum point: (1, -2), (-1, -2)

問題一：「6x(x^2 + 1)(x + 1)(x – 1) = 0」如何得出x = 0, 1 or -1？請列明計算步驟。
問題二：minimum point為何會有兩個，minimum不是最少的那個嗎？