What is Optimisation?
The process of finding the maximum or minimum value of a function, often subject to constraints. In calculus, this is done by analysing critical points found through differentiation.
Optimisation problems appear throughout mathematics and the real world. Whether a company wants to minimise costs, an engineer wants to maximise efficiency, or a farmer wants to maximise the area of a field — the mathematical approach is the same.
The key idea: if a function has a maximum or minimum on an open interval, it must occur at a critical point (where or is undefined). However, on a closed interval, you must also check the endpoints.
Think of optimisation like finding the highest or lowest point on a hiking trail between two fixed start and end points. The summit (maximum) or valley (minimum) could be somewhere in the middle — or it could be right at the start or end of the trail.