(6) - 10 pts. Let f(x) = 16x4 − 19x3 − x2 + 2 on [0,
(a) Use the Intermediate Value Theorem to show f has a real zero in
Solution: f(0) is positive and f (1) is negative, so f has a real zero in
(b) List all the possible zeros in the interval [0, 1].
Solution: p|2 and q|16, so p = ±1,±2 and q = ±1,±2,±4,±8,±16.
So the possible roots are ±1,±2,±1/2,±1/4,±1/8,±1/16,
but in (0, 1) we only have 1, 1/2, 1/4, 1/8, 1/16.
(7) - 10 pts. Let f be a complex polynomial of degree 6
with zeros: 7+i, −3+i, −i.
Determine the remaining zeros of f.
Solution: Negate the complex part of each root to yield
7 − i, −3 − i, i.
(8) - 30 pts. Dwight K. Schrute and his cousin, Mose, own
a beet farm, where they
grow both red and golden beets.
(a) Dwight and Mose have 7000 meters of fencing to enclose a rect-
angular area separating the red and golden beets. On one side of
the rectangular area is the driveway, which will not have fencing
along it(see Figure 1).
i. Express the area as a function of x.
Solution: The perimeter is given by x+2y = 7000 ! y = 1/2
x). Thus the area is
ii. Determine the maximum area that can be enclosed.
Solution: The area function is quadratic so the maximum occurs
So the maximum area is
A(3500) = 6125000.
(b) The price p and quantity x of a pallet of beets is
i. Find the revenue as a function of x.
ii. What quantity maximizes revenue?
Solution: Again, this is
iii. What is the maximum revenue?
Solution: R(206.25) = 13612.5
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