Download 4.4 Optimization I • absolute (global) extreme values Let f be a

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Mathematics 0110a Summary Notes
4.4
•
Optimization I
absolute (global) extreme values
Let f be a function with domain D. Then f(c) is the
(a) absolute maximum value on D if and only if f(x) ≤ f(c) for all x in D;
(b) absolute minimum value on D if and only if f(x) ≥ f(c) for all x in D.
These are also called the absolute extrema of f and are sometimes referred to as simply
the extrema or maxima and minima of f.
The existence of absolute extrema depends on both the function rule and domain.
Example 1:
function rule
domain
absolute extrema
1.
y = x2
(-∞, ∞)
absolute minimum = 0 at x = 0
no absolute maximum
2.
y = x2
[0, 4]
absolute minimum = 0 at x = 0
absolute maximum = 16 at x = 4
3.
y = x2
(0, 4]
no absolute minimum
absolute maximum = 16 at x = 4
4.
y = x2
(0, 4)
no absolute minimum
no absolute maximum
5.
y = x2
(-1, 4)
absolute minimum = 0 at x = 0
no absolute maximum
Notes: 1. If f is continuous on a closed interval then f has both a maximum and a
minimum on the interval. See 2 above.
2. If f is defined on an open interval then any extrema of f occur at interior
points of the interval (at points surrounded by other points of the interval).
See 5 above.
Mathematics 0110a Summary Notes
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Recall
•
relative extreme values
Let c be an interior point of the domain of the function f. Then f(c) is a
(a) relative maximum value at c if and only if f(x) ≤ f(c) for all x in some interval
containing c;
(b) relative minimum value at c if and only if f(x) ≥ f(c) for all x in some interval
containing c.
Notes: 1. Relative extrema are also called local extrema.
2. Relative extrema can also occur at the endpoints of closed intervals
provided the appropriate inequality holds.
3. Absolute extrema are also relative to any interval generating them.
•
finding extreme values
Relative extreme values: If a function f has a relative maximum value or a relative
minimum value at an interior point c of its domain, and if f ′ exists at c, then f ′ (c) = 0.
Notes: Relative and absolute extrema also exist at points in the domain where f ′ is
undefined (at corners or cusps).
A point (c, f(c)) in the domain of the function f at which f ′ = 0 or f ′ does not exist is a
critical point of f (and c is called a critical number of f).
Notes: 1. Extrema may occur only at critical numbers and endpoints of closed intervals.
2. Critical numbers are candidates for where f may have extrema.
More analysis is needed to determine whether or not a critical number
generates local extrema.
Procedure:
1. Find c in the domain of f such that either f ′ (c) = 0 or does not exist.
That is, find all the critical numbers of f.
2. Determine whether endpoints need be considered.
Yes, in the case of closed or half-open intervals.
3. Compare the values of f at all the points in 1., and 2., above.
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Mathematics 0110a Summary Notes
The largest value is the absolute maximum and the smallest value is
the absolute minimum.
Find any absolute extrema of f(x) = x2 − 6x +12 on the interval [1, 4].
Example 2:
solution: Step 1.
f ′(x) = 2x − 6 = 0 → x = 3 is the only critical number of f in [1, 4].
Step 2.
Absolute extrema are required on [1, 4] which is a closed interval and so its
endpoints need to be considered.
Step 3.
f(1) = 7
is the maximum value of f on [1, 4].
f(3) = 3
is the minimum value of f on [1, 4].
f(4) = 4
Example 3.
Let f (x) = x4 – 8x2.
Find the maximum and minimum value of f(x) on the interval [1, 3].
solution: Step 1.
f '(x) = 4x3 − 16x = 4x(x2 − 4)
= 4x(x + 2)(x – 2) .
So f has critical number 2 in [1, 3]. Note 0 and − 2 are excluded
because they don’t belong to the interval being considered.
Step 2.
[1, 3] is a closed interval. Its endpoints must be considered.
Step 3.
f(1) = −7
f(2) = −16
is the minimum value of f on [1, 3].
f(3) = 9
is the maximum value of f on [1, 3]