Question #41680

This question tests the student's ability to handle telescopic series combined with rational inequalities, specifically targeting the 'Wavy Curve' trap for even-multiplicity roots.

Students must first recognize the general term simplifies to a difference of squares, allowing the series to telescope.

The lethal collision occurs when solving the resulting inequality $\frac{12(x-5)}{(x-2)^2(x-8)^2} \ge 0$.

A common mistake is treating the roots of the denominator ($x=2, 8$) as points where the sign changes (odd multiplicity), leading to an incorrect feasible region of $x \in (2, 5)$.

In this fake region, the objective function $x+y$ reaches its unconstrained maximum of $6\sqrt{2}$.

However, correctly identifying that even-multiplicity roots do not change the sign reveals the true feasible region is $x \in (5, 6)$, where the function is strictly decreasing, making the supremum $5+\sqrt{11}$.