证明不等式急~~~证明不等式:lnx+1/x-1/2(x-1)^2>=1+2/3(1-x)^3希望大家帮助解决,有过程呦!

Question

证明不等式 急~~~
证明不等式:lnx +1/x-1/2(x-1)^2>=1+2/3(1-x)^3
希望大家帮助解决,有过程呦!

Answer 1

原命题等价于lnx+1/x-1/2(x-1)^2+2/3(x-1)^3≥1
构造f(x)=lnx+1/x-1/2(x-1)^2+2/3(x-1)^3,(x>0)
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