= Solution
The <Lagrange sufficiency theorem> says that for a convex differentiable objective and convex differentiable inequality constraints, any feasible point satisfying the <Karush-Kuhn-Tucker conditions> with nonnegative multipliers is a global minimizer.
Write
$$
f=-x_1-3x_2,\qquad
g_1=x_1^2+x_2^2-25,\qquad
g_2=-x_1+2x_2-5.
$$
The unconstrained maximizer of $x_1+3x_2$ on the disc violates $g_2\leq0$, so both boundaries are active at the optimum. Their intersections are
$$
(-5,0)\quad\text{and}\quad(3,4),
$$
and $(3,4)$ gives the smaller objective.
To certify it, at $x_*=(3,4)$ choose
$$
\lambda=\frac14,\qquad\mu=\frac12.
$$
Then
$$
\nabla f(x_*)+\lambda\nabla g_1(x_*)
+\mu\nabla g_2(x_*)=0,
$$
both multipliers are nonnegative, and complementary slackness holds because both constraints are active. The sufficiency theorem therefore gives
$$
\boxed{x_1=3,\qquad x_2=4,\qquad f_{\min}=-15}.
$$
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