Finding MLE of the mean and epsilon

Question

I want to find the MLE of $\mu$ and $\epsilon$, that are the parameters in the density function $$f(\epsilon,\mu) = (1-\epsilon)\frac{1}{3}e^{x/3}+\frac{\epsilon}{\mu}e^{x/\mu}.$$ I found the likelihood function L, $$L(\epsilon,\mu) = \Pi_i^n (1-\epsilon)\frac{1}{3}e^{x/3}+\frac{\epsilon}{\mu}e^{x/\mu} = \Big(\frac{1-\epsilon}{3}\Big)^n e^{\sum_i^n x_i/3}+\Big(\frac{\epsilon}{\mu}\Big)^n e^{\sum_i^n x_i/\mu}.$$ I am not sure if my above calculation is correct and what to do next. Thank you for your help!