Extrema represent the critical milestones in a function's journey. We distinguish between the Absolute (Global)—the ultimate peak or valley across the entire domain—and the Local—the peaks and valleys that are higher or lower than their immediate neighbors. These points are the primary targets when optimizing physical systems, from the trajectory of a rocket to the minimization of fuel consumption.
1. Formal Definitions of Extrema
Definition 1: Absolute Extrema
Let $c$ be a number in the domain $D$ of a function $f$.
- $f(c)$ is the absolute maximum if $f(c) \ge f(x)$ for all $x$ in $D$.
- $f(c)$ is the absolute minimum if $f(c) \le f(x)$ for all $x$ in $D$.
Definition 2: Local Extrema
$f(c)$ is a local maximum (or minimum) if $f(c) \ge f(x)$ (or $f(c) \le f(x)$) when $x$ is near $c$.
2. The Existence Guarantee: Extreme Value Theorem (EVT)
Finding a solution is only possible if a solution exists. The Extreme Value Theorem provides the guarantee: If $f$ is continuous on a closed interval $[a, b]$, then $f$ must attain both an absolute maximum and an absolute minimum.
Consider the contrast in transcendental functions:
- Example 1 (Periodic): $f(x) = \cos x$ reaches its absolute maximum of 1 infinitely many times (where $x = 2n\pi$).
- Example 3 (Power): $f(x) = x^3$ (on $(-\infty, \infty)$) has no extrema at all, as it increases and decreases boundlessly.
3. Symmetry and Growth
If $f(-x) = f(x)$, the function is even and symmetric about the $y$-axis. This implies that if a local minimum occurs at $x = 2$, an identical minimum must exist at $x = -2$. We see this in $f(x) = x^2$ (Example 2), where $f(0)=0$ is both a local and absolute minimum.
🎯 Core Principle
To find absolute extrema on $[a, b]$, evaluate the function at all critical numbers in the interior and at the endpoints $a$ and $b$. The largest value is the absolute max; the smallest is the absolute min.