A composite cylindrical rod of length \(2L\) consists of two segments of equal length \(L\) and equal cross-sectional area \(A\), made of Metal X (\(0 \le x \le L\)) and Metal Y (\(L \le x \le 2L\)), joined end-to-end. The lateral surfaces of both segments are thermally insulated. The left end (\(x = 0\)) is maintained at \(100\text{ }^\circ\text{C}\) and the right end (\(x = 2L\)) is maintained at \(10\text{ }^\circ\text{C}\). The system reaches a steady state, and the temperature \(T\) along the rod as a function of position \(x\) is shown in the graph. Which of the following correctly compares the thermal conductivities \(k_X\) and \(k_Y\) of the two metals, and provides a valid justification?