Global Extrema
Calculus & Advanced MathGlobal 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. They are the champions of all values, not just a local neighborhood.
Example
On a map of all mountains on Earth, Mount Everest is the global maximum elevation. The Mariana Trench is the global minimum.
Key Insight
A global extremum beats every other value in the whole domain, while a local extremum only beats nearby values.
Definition
The global (absolute) maximum of $f$ on $[a, b]$ is the largest value $f$ attains. By the Extreme Value Theorem, every continuous function on a closed interval achieves both a global maximum and minimum. To find them: evaluate $f$ at all critical points and at the endpoints.
Example
$f(x) = x^3 - 3x$ on $[-2, 2]$: $f'(x) = 3x^2 - 3 = 0$ at $x = \pm 1$. $f(-2) = -2$, $f(-1) = 2$, $f(1) = -2$, $f(2) = 2$. Global max $= 2$, global min $= -2$.
Key Insight
Always check endpoints! The global extremum on a closed interval often occurs at an endpoint, not at an interior critical point.
Definition
The Extreme Value Theorem: a continuous $f$ on a compact set $K \subset \mathbb{R}^n$ attains its global maximum and minimum. On open or unbounded domains, global extrema may not exist. Optimization problems seek global extrema subject to constraints, addressed by Lagrange multipliers or dynamic programming.
Example
$f(x) = x$ on $(0, 1)$: continuous but no global max or min (open interval). $f(x) = x^2$ on $\mathbb{R}$: global min at $0$ but no global max.
Key Insight
Compactness is the key structural condition ensuring global extrema exist. This motivates the practical technique of converting unconstrained problems to constrained ones over compact feasible sets.