= Solution
Use multipliers $\lambda,\mu\geq0$ for
$$
x_1+x_2-c\leq0,
\qquad
\sqrt{x_2}-d\leq0.
$$
At an interior point in the nonnegative quadrant, the <Karush-Kuhn-Tucker conditions> are
$$
\log x_1+1+\lambda=0,
$$
$$
-1+\lambda+\frac{\mu}{2\sqrt{x_2}}=0,
$$
with complementary slackness for the two constraints.
For $c=3/e^2$ and $d=2/e$, the square-root constraint is inactive at the answer, so $\mu=0$. The second stationarity equation gives $\lambda=1$, and the first gives $x_1=e^{-2}$. The sum constraint is active, hence
$$
\boxed{x_1=\frac1{e^2},
\qquad
x_2=\frac2{e^2}}.
$$
Indeed, $\sqrt{x_2}=\sqrt2/e<2/e$. The minimum value is
$$
\boxed{-\frac4{e^2}}.
$$
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