maximum value of function

Below is the graph for
y1 = x^3 – 3a^2x (red) focusing on the portion x >= 0
http://img508.imageshack.us/img508/1985/cubic.png


圖片參考：http://img508.imageshack.us/img508/1985/cubic.png


dy1/dx = 3x^2 – 3a^2
d^2y1/dx^2 = 6x
when dy1/dx = 0, x = a
d^2y1/dx^2 | x = a is 6a > 0 meaning minimum
minimum y(a) = a^3 – 3a^3 = -2a^3
y1 = 0 => x^3 – 3a^2x = 0
x(x^2 – 3a^2) = 0 => x = 0 or x = √3 a
For the curve y = | x^3 – 3a^2x | (blue portion is inverted from y1), there is a local maximum at x = a, and the maximum is 2a^3
The curve meets the x-axis at x = 0 and x = √3 a
The curve reaches y = 2a^3 at x = a as well as another point x = h such that 2a^3 = h^3 – 3a^2h
h^3 – 3a^2h – 2a^3 = 0
(h – a)^2(h – 2a) = 0
h = a (the local max point) or h = 2a
From the above, if a > 1, i.e. the maximum point is not included in the range [0,1], the maximum happens at the boundary, i.e. x = 1
And the maximum value is | 1 – 3a^2 | or 3a^2 – 1
If 1 > 2a, i.e. 1/2 > a, the maximum happens again at x = 1 and the value is 1 – 3a^2
If 1/2 <= a <= 1, the maximum = the local maximum at x = a. In summary, the maximum value is
{3a^2 -1 ; a > 1
{2a^3; 1/2 <= a <= 1{1 – 3a^2; 0 < a < 1/2

2010-01-17 23:22:45 補充：
http://img508.imageshack.us/img508/1985/cubic.png