Extreme Value Theorem, Global Versus Local Extrema, and Critical Points

A continuous curve drawn without lifting your pen, over a closed interval [a,b], cannot escape having a top and a bottom. That's the Extreme Value Theorem. The curve is trapped: it starts at f(a), ends at f(b), and never jumps or breaks in between, so it has to reach some highest point and some lowest point somewhere on that interval. Drop either condition, continuity or a closed interval, and the guarantee can fail.

The theorem tells you an extreme value exists, but not where. For that, you check critical points: places where f'(x) = 0 or where f'(x) doesn't exist. These are the only candidates for local extrema, because anywhere the derivative is positive or negative, the function is still climbing or falling, not turning around. On a closed interval, the absolute max and min each live either at a critical point or at one of the two endpoints, so you evaluate f at all of them and compare.

Finding a critical point is just step one. A zero derivative only means the tangent line is flat there; it doesn't say which way the curve turns next. That's why this topic sits right before the first and second derivative tests: those tests read the sign changes around each critical point to decide whether it's a max, a min, or neither. Global versus local matters too, since a local max is just the tallest point nearby, while the global max is the tallest point on the whole interval, and only the Extreme Value Theorem guarantees the global one exists at all.