Global Extrema
Global extrema are the absolute highest (global maximum) and absolute lowest (global minimum) values a function achieves over its entire domain or a closed interval.
Definition
The global maximum is the single highest point on an entire graph, and the global minimum is the single lowest point, the champions of all values rather than just a local neighborhood. The Extreme Value Theorem guarantees that every continuous function on a closed, bounded (compact) interval or set attains both a global maximum and minimum; to find them, evaluate $f$ at all critical points and at the endpoints. On open or unbounded domains, global extrema may not exist.
Example
On a map of all mountains on Earth, Mount Everest is the global maximum elevation and the Mariana Trench is the global minimum. For $f(x) = x^3 - 3x$ on $[-2, 2]$: $f'(x) = 3x^2 - 3 = 0$ at $x = \pm 1$; checking $f(-2) = -2$, $f(-1) = 2$, $f(1) = -2$, $f(2) = 2$ gives a global max of $2$ and global min of $-2$. By contrast, $f(x) = x$ on $(0, 1)$ is continuous but has no global max or min because the interval is open.
Key Insight
A global extremum beats every other value in the whole domain, while a local extremum only beats nearby values, so always check endpoints, the global extremum on a closed interval often occurs there rather than at an interior critical point. Compactness is the key structural condition ensuring global extrema exist, which motivates the practical technique of converting unconstrained problems to constrained ones over compact feasible sets, addressed by Lagrange multipliers or dynamic programming.