Final answer:
In the case of a CEO accused of questionable behavior, the Board of Directors should conduct an internal investigation to ensure any actions they take are based on substantiated facts. Other options like public accusations without evidence or ignoring the situation could be damaging to the company.
Step-by-step explanation:
If the CEO is accused of questionable behavior, the Board of Directors should take steps to ensure the integrity of the company is maintained. The most appropriate of the options listed is to conduct an internal investigation. This approach is responsible and allows the board to uncover facts before making any public statements or decisions. The investigation should be thorough, impartial, and conducted by individuals or an external party that does not have a conflict of interest.
Publicly accusing the CEO without evidence could lead to legal issues and harm the company's reputation. Ignoring the accusations could have similar effects if the allegations are later proven true. Requesting shareholders' opinion through a vote might not be the ideal first step before gathering facts, although shareholders should be kept informed as appropriate.