= Solution
{c}
For each fixed $\lambda>0$, maximizing the <likelihood function> over $\mu$ is equivalent to minimizing $A(\mu)=\sum_i|x_i-\mu|$. Order the distinct observations as $x_{(1)}<\cdots<x_{(n)}$ and write $n=2m+1$. Between successive <order statistics>, with exactly $k$ observations to the left,
$$
A'(\mu)=k-(n-k)=2k-n.
$$
Thus $A$ decreases up to $x_{(m+1)}$ and increases after it. Its unique <minimum> is at the <sample median>:
$$
\boxed{\widehat\mu=x_{(m+1)}.}
$$
At a fixed $\mu$ with $A(\mu)>0$, the <log-likelihood function> has <derivative> $n/\lambda-A(\mu)$ and second <derivative> $-n/\lambda^2<0$, so its unique maximizing rate is $n/A(\mu)$. The maximized <likelihood function> is $(n/[2eA(\mu)])^n$, which increases when $A(\mu)$ decreases. Hence the joint <maximum-likelihood estimates> are
$$
\boxed{\widehat\mu=x_{(m+1)},\qquad
\widehat\lambda=\frac{n}{\sum_i|x_i-x_{(m+1)}|}.}
$$
This is <Laplace location and scale maximum likelihood>, expressed with the reciprocal scale.
\b[A finite rate <maximum-likelihood estimate> requires a positive total absolute deviation.] For distinct observations and odd $n\ge3$, that condition holds. For $n=1$, however, $\mu=x_1$ gives $A(\mu)=0$ and $L=\lambda/2\to\infty$ as $\lambda\to\infty$. There is no finite joint <maximum-likelihood estimate> in the stated parameter space; the formula with zero denominator is not an admissible estimate.
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