Explain the necessary and sufficient conditions in case of unconstrained optimisation.

In the realm of optimization, whether it's in mathematics, engineering, economics, or any other field, the goal is to find the best possible solution, often represented by the maximum or minimum of a certain function. Unconstrained optimization refers to the process of finding the extremum of a function without any restrictions or constraints on the variables involved. The key to understanding unconstrained optimization lies in grasping the necessary and sufficient conditions that govern the existence and location of the extremum.

Necessary Conditions for Extremum:

Necessary conditions are criteria that must be satisfied for a point to be a potential extremum (maximum or ____ _________ _________ __________ ____ __________ _________ ____ ___ ______ ___ ________.
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