Optimization and Extrema

Finding the Best

Optimization — locating maxima and minima — is one of calculus's most powerful applications. Because a smooth function's tangent is horizontal at an interior peak or valley, the derivative is zero at these extreme points. This converts "find the maximum" into "solve $f'(x) = 0$."

The systematic method for a function on an interval:

1. Find the critical points — where $f'(x) = 0$ or where $f'$ is undefined.
2. Classify each as a local maximum, local minimum, or neither.
3. On a closed interval $[a, b]$, also evaluate the endpoints, since the extreme value may occur there even when the derivative is not zero.

Two tests classify a critical point:

• First-derivative test — examine the sign of $f'$ around the point: $+$ to $-$ marks a local maximum (rising then falling); $-$ to $+$ a local minimum.
• Second-derivative test — at a critical point, $f'' > 0$ gives a minimum (concave up), $f'' < 0$ a maximum (concave down); $f'' = 0$ is inconclusive.

The distinction between local and global (absolute) extrema matters: a local extremum is best only in its neighborhood; the global extremum is best over the whole domain. On a closed, bounded interval, the global max and min are found by comparing all critical-point values with the endpoint values — the largest and smallest among them.

Optimization is everywhere: minimizing cost or error, maximizing area or profit, finding the least-material design. Setting a derivative to zero to find the best choice is one of the most-used ideas in all of applied mathematics.

Common pitfall: assuming every point with $f'(x) = 0$ is a max or min, and forgetting the endpoints. A zero derivative marks a critical point, but it could be an inflection/saddle (like $x^3$ at 0, neither a max nor a min) — you must classify it. And on a closed interval the true (global) extreme may sit at an endpoint, where the derivative need not vanish; checking only $f'(x) = 0$ can miss the actual maximum.

A curve on a closed interval with accent horizontal tangents at interior peaks and valleys, plus the two endpoints flagged, and the tallest of all points circled as the global maximum.

$$f'(x) = 0 \text{ at interior extrema; check endpoints too}$$