Maximum Likelihood Estimation (absolute value)

Question

Im currently struggling on a problem. Let X1, . . . , Xn be a random sample of a random variable that has a pdf: $$f(x)=\frac{1}{2}\cdot e^{-\left |x-\theta \right |}, \infty <x<\infty $$ And I want to find the MLE for θ! So to solve this I start with finding the L(θ) for both when x>θ and when x<θ. For the following problem they will give the same answer so I derive L(θ) for only the first one, which gives us L(θ)=$2^{-n}\cdot e^{\sum_{k=1}^{n}-x_{k}+\theta }$. I then take the derivative with respect to θ of L(θ) and set it equal to zero. I get that $θ=\frac{1}{n}\cdot \sum_{k=1}^{n}x_{k}$ which is the sample mean. But according to the solution from my professor the answer should be that the MLE of θ is the median of the sample. So I guess I have done something wrong. I would really appreciate some help in how I should think. Thanks!