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.

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