= Solution
<Cramér's theorem> states that the means of <independent and identically distributed random variables> in $\mathbb R^d$ satisfy a <large deviation principle> at <large-deviation speed> $n$, with <good rate function> equal to the <Legendre-Fenchel transform> of their <cumulant-generating function>, provided the <moment-generating function> is finite on a neighborhood of zero.
Here the <exponential distribution> has
$$
\Lambda(\theta)=\log\frac{\lambda}{\lambda-\theta}\quad(\theta<\lambda),
\qquad \Lambda(\theta)=\infty\quad(\theta\ge\lambda).
$$
For $x>0$, maximize $\theta x-\Lambda(\theta)$. Its <derivative> is $x-(\lambda-\theta)^{-1}$ and its second <derivative> is negative, so the maximizing parameter is $\theta_x=\lambda-x^{-1}$. Therefore the <rate function of an exponential sample mean> is
$$
\boxed{I(x)=\begin{cases}\lambda x-1-\log(\lambda x),&x>0,\\+\infty,&x\le0.\end{cases}}
$$
For $x\le0$, letting $\theta\to-\infty$ in the same supremum gives $+\infty$, including the logarithmic divergence when $x=0$. The <moment-generating function> is finite near zero, so <Cramér's theorem> supplies the full <large deviation principle>.
The <rate function> is nonnegative, with its unique minimum at $\mu=\lambda^{-1}$; $I''(x)=x^{-2}>0$. It tends to infinity both as $x\downarrow0$ and as $x\to\infty$. Its finite <sublevel sets> are closed bounded intervals contained in $(0,\infty)$, hence <compact>. Thus it is a <good rate function>.
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