'Ashtadhyayi', the earliest book on sanskrit grammar was written by ____.

Panini

The correct answer is Option (3) → Panini

'Ashtadhyayi', the earliest book on sanskrit grammar was written by PANINI.

Ashtadhyayi of Panini was composed around c. 500 BCE. It is a work on Sanskrit grammar.

Some other major compositions:

Natyashastra of Bharata, a work on dramaturgy, was composed in Sanskrit around c. 300 CE.
Manusmriti is a text containing the various functions to be carried out by different Varnas among other things.
Charaka and Sushruta Samhitas are works on medicine (in Sanskrit).