Question Number 97983 by abdomathmax last updated on 10/Jun/20
![f continue on [0,1] and f(x)>0 on [0,1] prove that ∫_0 ^1 lnf(x)dx≤ln(∫_0 ^1 f(x)dx)](https://www.tinkutara.com/question/Q97983.png)
$$\mathrm{f}\:\mathrm{continue}\:\:\mathrm{on}\:\left[\mathrm{0},\mathrm{1}\right]\:\mathrm{and}\:\mathrm{f}\left(\mathrm{x}\right)>\mathrm{0}\:\mathrm{on}\:\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{lnf}\left(\mathrm{x}\right)\mathrm{dx}\leqslant\mathrm{ln}\left(\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\right) \\ $$