Consider $X_1,X_2,...,X_n$ a random sample of an statistic population that is modeled by the density function:$$f(x)= \frac{1}{\lambda}e^{-(X-\delta)/\lambda} I_{[\delta,\infty)}$$
Obtain maximum likelihood estimators for $\delta $ and $\lambda$
We know $$L(\delta,\lambda;X)=\prod_{i=1}^N \frac{1}{\lambda}e^{-(X-\delta)/\lambda}=\frac{1}{\lambda^n}e^{-\frac{1}{\lambda}(\sum_{i=1}^NX_i-n\delta)}$$
Taking the log likelihood funtion we get
$$l(\delta,\lambda;X)=-n\ln (\lambda)-\frac{1}{\lambda}\sum_{i=1}^N (X_i-n\delta)$$
Taking the derivative with respect to the each parameter
$$\frac{\partial l}{\partial \delta}=\frac{n}{\lambda}$$
and
$$\frac{\partial l}{\partial \lambda}=-\frac{n}{\lambda}+(\sum_{i=1}^N X_i-n\delta)(\frac{1}{\lambda^2})$$
My problem comes when I want to maximize the functions. For $\hat\delta$, I think I need to involve minimum values; for $\frac{\partial l}{\partial \delta}=0$, I am not sure what to do. How can I proceed?