令 h(x) = g(x) - f(x) = 2/3x³ - 1/2x² - lnx
h(1) = 2/3 - 1/2 = 1/6 > 0
表明在x=1处,g(x)的图像在 f(x)的上方.
dh/dx = 2x² - x - 1/x
= [2x³ - x² - 1]/x
= [2x³ - 2x² + x² - 1]/x
= [2x² (x - 1) + (x-1)(x+1)]/x
= (x - 1)(2x² + x + 1)]/x
when x > 1,dh/dx > 0
这表示 h(x) 是严格递增函数,即：距离越来越大.
也就是说,从x〉1开始,g(x)的图像永远在f(x)的图像之上.