What is the maximum and minimum values of a function in a closed interval
Answers
Find maximum and minimum values of a function over a closed interval
Facts: Let f(x) be a function on [a, b] and c is a point in the interval [a, b].
(1) If for any point x in [a, b], f(x) ≥ f(c) (respectively, f(x) ≤ f(c)), then f(c) is the
absolute (or global) minimum value (respectively, absolute (or global) local max-
imum value) of f(x) on [a, b].
(2) If a < c < b, and for any point x in an open interval containing c, f(x) ≥ f(c) (re-
spectively, f(x) ≤ f(c)), then f(c) is a local minimum value of f(x) (respectively, local
maximum value) on [a, b].
(3) If f(x) is continuous on [a, b] and differentiable in (a, b), a point c in [a, b] is a critical
point of f(x) if either f
0
(c) does not exist, or f
0
(x) = 0.
(4) Important: If f(x) is continuous on [a, b] and differentiable in (a, b), and if for some c
in (a, b), f(c) is a local maximum or local minimum, then c must be a critical point. Any
absolute maximum or minimum must take place at critical points inside the interval or at
the boundaries point a or b.
Example 1 State whether the function f(x) = |x − 2| attains a maximum value or a
minimum value in the interval (1, 4].
Solution: Apply the definition of absolute value to get
f(x) =
x − 2 if 2 ≤ x ≤ 4,
2 − x if 1 < x < 2.
Thus the graph of this function consists of two pieces of lines, and so the minimum value
f(2) = 0 @ x = 2, and the maximum value is f(4) = 2 @ x = 4.
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Extreme value theorem tells us that a continuous function must obtain absolute minimum and maximum values on a closed interval. These extreme values are obtained, either on a relative extremum point within the interval, or on the endpoints of the interval.