求下列式子的极值 (1) 5x-3x^2 (2) 2x^2+x-3 (3) 1/(x^2+x+1) (4) x^2+4/x^2 感激不尽
问题描述：
求下列式子的极值 (1) 5x-3x^2 (2) 2x^2+x-3 (3) 1/(x^2+x+1) (4) x^2+4/x^2 感激不尽
答
(1)
-3x^2 + 5x = -3 (x-5/6)^2 + 25/12
当x= 5/6时有极大值25/12
(2)
2x^2 + x - 3 = 2(x+1/4)^2 - 25/8
当x=-1/4时有极小值-25/8
(3)
因为x^2 + x + 1 = (x+1/2)^2 + 3/4
所以1/(x^2 + x + 1)在x=-1/2时有极大值4/3
(4)
x^2 + 4/x^2 >= 2*x*(2/x) = 4
当x^2 = 4/x^2,也即x=正负根号2时,有极小值4