MLE problem - the likelihood function has no maximum. The probability density function is:
$f(x)=e^{\theta -x}, \ 0 \le \theta \le x $
Given an n-element sample, the likelihood function is:
$$L(\theta)=\exp \left( n\theta - \sum_{i=1}^n x_i \right)$$
Since the function has no maximum, but increases with $\theta$, I would take the estimator to be $\hat{\theta}=\max X_i$, following an example I did in class. But the answers I have (not 100% correct) say to take the minimum. Can someone shed some light on this?

This will be clear if you write the likelihood function more carefully as
\begin{align}
L(\theta) &= e^{  \left( n\theta - \sum_{i=1}^n x_i \right) }\prod_{i=1}^n 1_{\{ \theta \le x_i\}} \\
&= e^{  \left( n\theta - \sum_{i=1}^n x_i \right) }1_{\{ \theta \le \min\limits_{1\le i\le n} x_i\}}
\end{align}
As you correctly noted, $L(\theta)$ is an increasing function of $\theta$ on the interval $[0, \min x_i]$. And for $\theta > \min x_i$, the likelihood is zero. 
Thus, $\hat{\theta} = \min X_i$.
