Optimization and Extrema

Finding the Best

Optimization—finding maxima and minima—turns shape into strategy. Because a smooth function's tangent is horizontal at an interior peak or valley, its derivative is zero at extreme points. This reduces optimization to solving $f'(x) = 0$.

Common Pitfall: Never assume $f'(x) = 0$ guarantees a max or min. It may be a flat inflection point (like $x^3$ at 0). Always test the point, and never forget endpoints, where the global extreme often hides!

Classifying Extrema

Two primary tools classify a critical point:

• First-derivative test: Watch the sign of $f'$ around the point. Plus to minus ($+$ to $-$) is a local maximum (rising then falling); minus to plus ($-$ to $+$) is a local minimum.
• Second-derivative test: At a critical point, $f''(x) > 0$ means local minimum (concave up), and $f''(x) < 0$ means local maximum (concave down). If $f''(x) = 0$, the test is inconclusive.

Keep the scale in mind: a local extremum is best only in its immediate neighborhood, while a global extremum is the absolute best across the entire domain. On a closed interval $[a, b]$, find the global max and min by evaluating every critical point and endpoint, then taking the largest and smallest values.