The correct answer is: A) Banavasi.
The Kadamba dynasty was a Kannada dynasty that ruled in the Deccan region of India from the 3rd to the 6th centuries CE. The dynasty was founded by Mayura Varman I in 345 CE, and its capital was at Banavasi in the present-day state of Karnataka. The Kadamba dynasty was the first indigenous Kannada-speaking kingdom, and it played a significant role in the development of Kannada culture and literature.
The Kadamba dynasty was founded by Mayura Varman I, who was a feudatory of the Satavahana dynasty. Mayura Varman I declared independence from the Satavahanas in 345 CE, and he established the Kadamba dynasty. The Kadamba dynasty reached its zenith under the rule of Harivarman I (455-500 CE), who conquered a large part of the Deccan region. Harivarman I was a great patron of art and literature, and he built many temples and monasteries.
The Kadamba dynasty declined after the death of Harivarman I, and it was eventually overthrown by the Chalukya dynasty in the 6th century CE. However, the Kadamba dynasty left a lasting legacy on the Deccan region, and it is considered to be one of the most important dynasties in the history of Karnataka.
The other options are incorrect because they were not the capitals of the Kadamba dynasty. Badami was the capital of the Chalukya dynasty, Aihole was a major center of Chalukya art and architecture, and Manyakheta was the capital of the Rashtrakuta dynasty.