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
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– 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.
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