= Solution
For an observed $x\ge0$, the <continuous uniform distribution> likelihood is $p(x\mid\theta)=\theta^{-1}1_{\{\theta\ge x\}}$, with $\theta>0$. Multiplying by the stated prior cancels the factor $\theta$, giving $e^{-\theta}1_{\{\theta\ge x\}}$. Its integral over the permitted parameter values is $e^{-x}$. Hence the <posterior distribution> is
$$
\boxed{p(\theta\mid x)=e^{-(\theta-x)}1_{\{\theta\ge x\}},\qquad \theta\mid x\sim x+\operatorname{Exp}(1).}
$$
This is the <exponential posterior for a uniform endpoint>. Its posterior mean is $x+1$ and its variance is one. The posterior expected <squared-error loss> decomposes as
$$
E[c(\theta-a)^2\mid x]=c\operatorname{Var}(\theta\mid x)+c(a-E[\theta\mid x])^2=c+c(a-x-1)^2.
$$
Since $c>0$, its unique minimum occurs at \b[the Bayes estimate] $\boxed{\widehat\theta_B=x+1}$. The positive loss multiplier changes the risk, not the optimizing action.
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