Extrema On An Interval

Understanding Extrema on Closed Intervals

When finding the largest and smallest values of a function on a closed interval $[a,b]$, you're locating absolute extrema. The Extreme Value Theorem guarantees these extrema exist whenever $f$ is continuous on a closed interval — though it doesn't tell you where to find them.

Absolute vs. Local Extrema

An absolute maximum is the single largest value over the entire interval; an absolute minimum is the smallest. A local extremum is simply the largest or smallest value near a particular point. Every local extremum in the interior is a candidate for an absolute extremum, but absolute extrema can also occur at endpoints (and may not be local extrema).

Finding Critical Numbers

Extrema can only occur at:

• Points where $f'(c) = 0$, or
• Points where $f'(c)$ is undefined (but $c$ remains in the domain)

These are called critical numbers. Not every critical number is an extremum — for instance, $f(x) = x^3$ has a critical number at $x=0$ but no local extremum there (the derivative doesn't change sign).

The Closed Interval Method

To find absolute extrema on $[a,b]$:

1. Find all critical numbers in the open interval $(a,b)$
2. Evaluate $f$ at each critical number and at both endpoints $a$ and $b$
3. The largest value is the absolute maximum; the smallest is the absolute minimum

Example: For $f(x) = x^3 - 3x$ on $[-2, 3]$, we have $f'(x) = 3x^2 - 3$, giving critical numbers $x = -1$ and $x = 1$. Evaluating: $f(-2) = -2$, $f(-1) = 2$, $f(1) = -2$, $f(3) = 18$. The absolute maximum is $18$ at $x=3$, and the absolute minimum is $-2$ (at both $x=-2$ and $x=1$).

Key Reminder: Always check where derivatives are undefined—corners, cusps, and vertical tangents matter just as much as flat tangent lines when hunting for extrema.