求证：lnx+1/x-1/2(x-1)2>=1+2/3(1-x)3,(x>0)

求证：lnx+1/x-1/2(x-1)2>=1+2/3(1-x)3,(x>0)
求证：lnx+1/x-1/2(x-1)2>=1+2/3(1-x)3,(x>0)

求证：lnx+1/x-1/2(x-1)2>=1+2/3(1-x)3,(x>0)
原命题等价于lnx+1/x-1/2(x-1)^2+2/3(x-1)^3≥1(x>0)
构造f(x)=lnx+1/x-1/2(x-1)^2+2/3(x-1)^3
f'(x)=1/x-1/x^2-(x-1)+2(x-1)^2=(2x+1)(x-1)^3/x^2
令f'(x)≥0得到x≥1,当0