maxima and minima

Think of a stock price over a single day. There are little bumps where it is higher than the moments just around it — those are local highs — and the single greatest price it touched all day, the global high. Lows work the same way. Maxima and minima are simply the high and low points of a function, and calculus gives us a reliable way to hunt them down instead of squinting at a graph.

We distinguish local (or relative) extremes from global (or absolute) ones. A local maximum is a point higher than everything nearby; the global maximum is the highest value over the whole domain considered. To classify a critical point, the first-derivative test checks whether f' switches from positive to negative (a local max), from negative to positive (a local min), or keeps the same sign (neither). The second-derivative test is often quicker: if f'(c) = 0 and f''(c) < 0 the curve is concave down so c is a local max; if f''(c) > 0 it is concave up so c is a local min; if f''(c) = 0 the test is inconclusive.

It is worth keeping two ideas separate: every global extreme on a closed interval is also a local extreme or an endpoint, but a local extreme need not be global — a small hill can sit far below the mountain. The Extreme Value Theorem guarantees that a continuous function on a closed, bounded interval actually attains both a global max and a global min; without continuity or without closing the interval, that guarantee can fail.