Final answer:
The X-bar and R-chart are quality management tools used to detect overstatements like Hyundai and Kia's fuel efficiency claims, and such an 'inconvenience' is categorized as an External failure cost.
Step-by-step explanation:
The quality management tool useful to detect overstatement like the fuel efficiency claims made by Hyundai and Kia is the X-bar chart (X-chart) in companion with the Range chart (R-chart). This tool is appropriate because it would monitor process performance over time to detect such deviations from claimed performance metrics. The category of cost of quality this 'inconvenience' corresponds to is External failure cost, because the cost is associated with defects that are found after the product reaches the customer. In this case, Hyundai and Kia had to compensate customers for the additional fuel costs incurred due to their overstatement of fuel efficiency, plus an additional 15% 'inconvenience' fee, representing a direct financial impact due to failure to meet quality expectations after delivery.