To determine which statements are correct regarding the authority of local governments in India, we need to understand the constitutional framework and the relationship between state and local governments.

1. Local government derived authority from the state government and are subordinates to it: This statement is partially correct. Local governments have a level of dependency on state governments for certain administrative and financial purposes, but they are not purely subordinate entities. They have constitutional status which grants them a degree of autonomy.
2. Local government are dependent on the state government: While local governments do rely on state governments for funding and certain regulatory oversight, their fundamental authority is derived from the Constitution, which allows them a level of independence for their local governance functions.
3. Local Government derived their authority from the constitution: This statement is correct. The 73rd and 74th Constitutional Amendments (1992) provided constitutional status to Panchayati Raj Institutions and Urban Local Bodies, respectively. This means their authority is derived directly from the Constitution, granting them significant autonomy in governance.
4. Local government perform function in the interest of the nation: While local governments indeed aim to serve the broader interests of the country, their primary role is to address local issues and needs. Therefore, while they indirectly contribute to national interests, they primarily focus on local governance and development.

Upon analyzing these statements, option Local Government derived their authority from the constitution is the most accurate because it correctly reflects the legal standing of local governments in India.