Statistics help

1) For each of the following, determine the intervals on which the following functions are increasing and decreasing and classify each of the critical points as a relative minimum, a relative maximum, or neither one.

a) 𝑓(𝑥) = 4𝑥5 − 95𝑥4 + 400𝑥3

1b) 𝑔(𝑥) = 𝑥 2 − 8

𝑥 + 3

1c) ℎ(𝑥) = 3𝑥(2𝑥 − 21)4/3

2) For each of the following, determine the intervals on which the following functions are concave up and concave down.

a) 𝑓(𝑥) = (3𝑥 − 18)𝑒𝑥

2b) 𝑔(𝑥) = 2 5 𝑥6 − 5𝑥4 − 216𝑥2

3) An apple grower finds that if she plants 36 trees per acre, each tree will yield 80 bushels of apples. She also estimates that for each additional tree that she plants per acre, the yield of each tree will decrease by 2 bushels. How many trees should she plant per acre to maximize her harvest and what is the maximum harvest? 4) Find the absolute extreme values of 𝑓(𝑥) = 𝑥4 − 4𝑥3 − 20𝑥2 on the interval [−3, 4].

5) A homeowner wants to enclose a 5,400 square foot rectangular garden by a fence in his backyard. If 3 sides of the fence cost $5.00 per foot and the 4th side costs $7.00 per foot, find the dimensions that will minimize the cost of building the fence and the minimum cost of its construction.

6) After t hours on a particular day, a freight train is 𝑠(𝑡) = −16.58𝑡 + 16.29𝑡2 − 0.76𝑡3 miles due north of its starting point (for 0 ≤ 𝑡 ≤ 18).

a) Where is the train (both distance and direction from starting point) after 11 hours?

b) What is the velocity of the train after 11 hours?

c) What is the acceleration of the train after 11 hours?

d) Is the train speeding up or slowing down after 11 hours? How can you tell?

7) Suppose that a company decides to raise capital by selling stock. Over the next 15 years the average monthly price of the stock fluctuates according to the rule 𝑆(𝑡) = 8.4𝑡0.75 − 0.35𝑡 + 32.15 where 𝑆(𝑡) is in dollars per share and t is the number of months since the stock was first offered for sale (this means that 𝑆(𝑡) is only valid on the interval [0, 180] ). Determine the maximum and minimum prices of the stock (in dollars and cents) and when these prices occurred.