Currently, I have a likelihood function in the following form:
$$L(\lambda|\mathbf{x})=\prod_{i=1}^{n}(x_i+(-1)^{x_i}e^{-\lambda})$$
with $x_i$ taking on values of $0$ or $1$, $\lambda>0$.
I have tried taking $log$ and differentiating the likelihood function, but have been unable of finding the maximum likelihood estimate of $\lambda$, as the result becomes
$$\frac\partial{\partial\lambda}log(L(\lambda|\mathbf{x}))=\sum_{i=1}^n\frac{-1}{1-x_ie^\lambda}$$
Can anyone give me some hints on how to do that?