How to prove the probability inequation with expectation?

Question

$X_1$ takes values in $[0,1]$. Let $EX_1=p$. Prove:

$$E(e^{tX_1})\le e^{(e^t-1)p}$$

The Markov Inequality gives the contrary sign and MGF seems not work. Can I directly derive this inequation? Much thanks!