190
3.3. GLOBAL OPTIMIZATION
any critical numbers mentioned in the problem statement represent all of the critical
numbers the function has in [a, b]. In each case, write one sentence to explain your
answer.
(a) f ′ (x) ≤ 0 for all x in [a, b]
(b) g has a critical number at c such that a < c < b and g ′ (x) > 0 for x < c and
g ′ (x) < 0 for x > c
(c) h(a) = h(b) and h ′′ (x) < 0 for all x in [a, b]
(d) p(a) > 0, p(b) < 0, and for the critical number c such that a < c < b, p ′ (x) < 0
for x < c and p ′ (x) > 0 for x > c
4. Let s(t) = 3 sin(2(t −
π
6 )) + 5. Find the exact absolute maximum and minimum of s on
the provided intervals by testing the endpoints and finding and evaluating all relevant
critical numbers of s.
(a) [
π
6 ,
7π
6 ]
(b) [0,
π
2 ]
(c) [0, 2π]
(d) [
π
3 ,
5π
6 ]