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Extrema and critical numbers Math 110 Let f be a function on some domain (for example on an interval [a, b] or (a, ∞) or even the whole real line (−∞, ∞)). Definitions. • A number x in the domain of f is a local minimum (or local min) of f if f (x) ≤ f (y) for all y near to x (in the domain of f ). • A number x in the domain of f is a local maximum (or local max ) of f if f (x) ≥ f (y) for all y near to x (in the domain of f ). • A number x in the domain of f is a global minimum (or global min) of f if f (x) ≤ f (y) for all y in the domain of f . • A number x in the domain of f is a global maximum (or global max ) of f if f (x) ≥ f (y) for all y in the domain of f . • A number x in the domain of f is an extreme point (or extremum) of f if it is a local minimum or a local maximum of f . • A number x in the domain of f is a critical number if either 1 – f 0 (x) = 0, – f 0 (x) is undefined, or – x is an endpoint of the domain of f . Note. • Every global maximum is a local maximum. • Every global minimum is a local minimum. • A function need not have any local maxima or minima (e.g. f (x) = x on the domain (−∞, ∞)). • A local or global maximum can also be a local or global minimum (e.g. every point in the domain of f (x) = 1 is a global maximum and minimum). • Critical numbers are the points where a min/max may occur. The extreme value theorem. If f is a continuous function on a closed interval [a, b], then f attains its local maximum and minimum values. 2