Final answer:
The best diagnostic test for a 40-year-old man with gradually progressive dementia and abnormal involuntary movements and a family history of similar illness would be a clinical evaluation and a thorough medical history.
Step-by-step explanation:
The best diagnostic test for a 40-year-old man with gradually progressive dementia and abnormal involuntary movements, along with a family history of similar illness, would be a clinical evaluation and a thorough medical history. Given the symptoms and the family history, Parkinson's disease is a likely diagnosis. A neurologist may order additional tests, such as brain imaging (MRI or CT scan) to rule out other possible causes and confirm the diagnosis.