Obtaining the correct Log-likelihood function $X_1, ..., X_n$ is a random sample from a population with pdf given by
$$
f(x; \mu, \lambda) = \frac{\lambda}{2}\operatorname{exp}(- \lambda |x - \mu|)
$$
where $\mu \in \mathbb{R}$ is the location parameter, and $\lambda > 0$ is the scale parameter.
I'm trying to obtain the log-likehood function to this problem. I have,
$$
L(\mu, \lambda; \mathbf{x}) = \prod_{i=1}^{n} f(x_i; \mu, \lambda) = \prod_{i=1}^{n} \frac{\lambda}{2}\operatorname{exp}(- \lambda |x - \mu|) = \frac{\lambda^n}{2^n}\operatorname{exp}\Bigl(- \lambda \sum_{i=1}^{n} |x - \mu|\Bigr).
$$
Let $ \ell(\mu, \lambda; \mathbf{x}) = \text{log}[L(\mu, \lambda; \mathbf{x})] $. Then we have
$$
\ell(\mu, \lambda; \mathbf{x}) = n\text{log}(\lambda/2) - \lambda \sum_{i=1}^{n} |x - \mu|.
$$
Is this the correct form of the sample log-likelihood? Because if it is... then trying to maximize it results in a Hessian determinant which is negative, even before evaluation at the critical point.

Assuming the log-likehood function is correct, you'd get:
$$
\frac{\partial}{\partial \mu} \ell(\mu, \lambda; \mathbf{x}) = \lambda \sum_{i=1}^{n} \frac{x_i - \mu}{|x_i - \mu|}
$$
$$
\frac{\partial}{\partial \lambda} \ell(\mu, \lambda; \mathbf{x}) = \frac{n}{\lambda} - \sum_{i=1}^{n} |x_i - \mu|
$$
$$
\frac{\partial^2}{\partial \lambda^2} \ell(\mu, \lambda; \mathbf{x}) = \frac{-n}{\lambda^2}.
$$
$$
\frac{\partial^2}{\partial \mu^2} \ell(\mu, \lambda; \mathbf{x}) = 0.
$$
$$
\frac{\partial^2}{\partial\lambda \partial \mu} \ell(\mu, \lambda; \mathbf{x}) = \frac{\partial^2}{\partial\mu \partial \lambda} \ell(\mu, \lambda; \mathbf{x}) = \sum_{i=1}^{n} \frac{x_i - \mu}{|x_i - \mu|}.
$$
Also I got the (single) critical point $ (\mu, \lambda) = (\bar{x}, n/\sum_{i=1}^{n} |x_i - \mu|) $. With this information you'd get $\text{det}(Hessian) < 0 \Rightarrow $ saddle point. So there's something wrong.
 A: The loglikelihood is correct. Your problem is the derivatives.
The loglikelihood is not differentiable with respect to $\mu$ where $\mu=x_i$ (and in particular, not at the MLE). The second derivative with respect to $\mu$ is zero everywhere that it is defined. So you can't just do calculus.
For any fixed $\lambda$, the loglikelihood is piecewise linear (and concave) in $\mu$ with corners at the observations, so there is a maximum (and no other stationary points) and the maximum must be at an observation. It's fairly easy to satisfy yourself that the maximum is at the median if $n$ is odd and on the whole median interval if $n$ is even.  That's true for every fixed $\lambda$ so it must be true for varying $\lambda$ as well.
Now, fixing $\mu$ at the MLE you have a differentiable one-parameter problem in $\lambda$. Solving
$$\frac{\partial}{\partial \lambda} \ell(\hat\mu, \lambda ; \mathbf{x})=\frac{n}{\lambda}-\sum_{i=1}^{n}\left|x_{i}-\hat\mu\right|$$
gives you $\hat\lambda$ as the mean absolute deviation from the median as the only stationary point. You can then check the second derivative wrt $\theta$ to make sure it's a maximum. Or you can argue that the maximum must be there or at the endpoints $\lambda=0$ or $\lambda=\infty$, and it's not either of those.
