Maximize an Integral involving a probability distribution

Calculus Level 5

Let \(p:[0, \infty)\to (0,\infty)\) be a continuous function such that \[\int_{0}^{\infty}p(x) \, dx=1\] Over all such functions \(p(x)\), define the integral \(I_p\) as follows \[I_p= \int_{0}^{\infty} e^{-4x}\ln(p(x)) \, dx\] Find \[\max_p I_p\]

Hint: The inequality \(e^x \geq 1+x\) might come handy.


This problem can be solved by using knowledge of high-school calculus only. Good luck!
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