3.3. GLOBAL OPTIMIZATION
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of the function or a particular interval.
• In the case of finding global extremes over the function’s entire domain, we again use
a first or second derivative sign chart in an effort to make overall conclusions about
whether or not the function can have a absolute maximum or minimum. If we are
working to find absolute extremes on a restricted interval, then we first identify all
critical numbers of the function that lie in the interval.
• For a continuous function on a closed, bounded interval, the only possible points at
which absolute extreme values occur are the critical numbers and the endpoints. Thus,
to find said absolute extremes, we simply evaluate the function at each endpoint and
each critical number in the interval, and then we compare the results to decide which
is largest (the absolute maximum) and which is smallest (the absolute minimum).
Exercises
1. Based on the given information about each function, decide whether the function
has global maximum, a global minimum, neither, both, or that it is not possible to
say without more information. Assume that each function is twice differentiable and
defined for all real numbers, unless noted otherwise. In each case, write one sentence
to explain your conclusion.
(a) f is a function such that f ′′ (x) < 0 for every x.
(b) g is a function with two critical numbers a and b (where a < b), and g ′ (x) < 0
for x < a, g ′ (x) < 0 for a < x < b, and g ′ (x) > 0 for x > b.
(c) h is a function with two critical numbers a and b (where a < b), and h ′ (x) < 0
for x < a, h ′ (x) > 0 for a < x < b, and h ′ (x) < 0 for x > b. In addition,
lim x→∞ h(x) = 0 and lim x→−∞ h(x) = 0.
(d) p is a function differentiable everywhere except at x = a and p ′′ (x) > 0 for
x < a and p ′′ (x) < 0 for x > a.
2. For each family of functions that depends on one or more parameters, determine the
function’s absolute maximum and absolute minimum on the given interval.
(a) p(x) = x 3 − a 2 x, [0, a] (a > 0)
(b) r(x) = axe −bx , [
1
2b , b] (a, b > 0)
(c) w(x) = a(1 − e −bx ), [b, 3b] (a, b > 0)
(d) s(x) = sin(k x), [
π
3k ,
5π
6k ]
3. For each of the functions described below (each continuous on [a, b]), state the location
of the function’s absolute maximum and absolute minimum on the interval [a, b], or
say there is not enough information provided to make a conclusion. Assume that
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of the function or a particular interval.
• In the case of finding global extremes over the function’s entire domain, we again use
a first or second derivative sign chart in an effort to make overall conclusions about
whether or not the function can have a absolute maximum or minimum. If we are
working to find absolute extremes on a restricted interval, then we first identify all
critical numbers of the function that lie in the interval.
• For a continuous function on a closed, bounded interval, the only possible points at
which absolute extreme values occur are the critical numbers and the endpoints. Thus,
to find said absolute extremes, we simply evaluate the function at each endpoint and
each critical number in the interval, and then we compare the results to decide which
is largest (the absolute maximum) and which is smallest (the absolute minimum).
Exercises
1. Based on the given information about each function, decide whether the function
has global maximum, a global minimum, neither, both, or that it is not possible to
say without more information. Assume that each function is twice differentiable and
defined for all real numbers, unless noted otherwise. In each case, write one sentence
to explain your conclusion.
(a) f is a function such that f ′′ (x) < 0 for every x.
(b) g is a function with two critical numbers a and b (where a < b), and g ′ (x) < 0
for x < a, g ′ (x) < 0 for a < x < b, and g ′ (x) > 0 for x > b.
(c) h is a function with two critical numbers a and b (where a < b), and h ′ (x) < 0
for x < a, h ′ (x) > 0 for a < x < b, and h ′ (x) < 0 for x > b. In addition,
lim x→∞ h(x) = 0 and lim x→−∞ h(x) = 0.
(d) p is a function differentiable everywhere except at x = a and p ′′ (x) > 0 for
x < a and p ′′ (x) < 0 for x > a.
2. For each family of functions that depends on one or more parameters, determine the
function’s absolute maximum and absolute minimum on the given interval.
(a) p(x) = x 3 − a 2 x, [0, a] (a > 0)
(b) r(x) = axe −bx , [
1
2b , b] (a, b > 0)
(c) w(x) = a(1 − e −bx ), [b, 3b] (a, b > 0)
(d) s(x) = sin(k x), [
π
3k ,
5π
6k ]
3. For each of the functions described below (each continuous on [a, b]), state the location
of the function’s absolute maximum and absolute minimum on the interval [a, b], or
say there is not enough information provided to make a conclusion. Assume that