chapter 7 hw and solutions

Use the Second Derivative Test to locate any relative extrema, if they exist, for the function f(x)=1/x^2-3

f has a relative maximum of - 1/3 at x = 0

A real estate firm owns 70 apartments. At $200 per month each apartment can be rented. However, for each $10 per month increase, two fewer apartments are rented. Determine the rent per apartment that will maximize monthly revenue.

$275

A drug that stimulates reproduction is introduced into a colony of bacteria. After t minutes, the number of bacteria is given approximately by N(t) = 1000 + 36t^2 - t^3, 0 ≤ t ≤ 30 How many minutes after the drug has been introduced is the number of bacteria at a maximum? What is the maximum number of bacteria?

24 minutes; 7912 bacteria

An open box is to be made by cutting equal squares from each corner of a 28-inch square piece of cardboard and then folding up the sides. Find the length of the side of the square that must be cut out if the volume of the box is to be maximized.

4 2/3 inches

From past records, the owner of Motel Morpheus has determined that when $x per day is charged to rent a room the daily profit, P(x), is given by P(x) = -x^2 + 76x - 100, 30 ≤ x ≤ 60 What should the owner charge to maximize profit?

The owner should charge $38 to maximize the profit.

A closed box is to be made with a volume of 512 cubic inches. The base of the box is to be a square. The material for the sides and top cost $0.02 per square inch, and the material for the base costs $0.04 per square inch. Find the dimensions of the box that minimize the cost of the materials.

base ≈ 6.99 inches by 6.99 inches height ≈ 10.48 inches

Use the Second Derivative Test to locate any relative extrema, if they exist, for the function f(x)=x^3+9/2x^2-12x-5

f has a relative maximum of 51 at x = -4 f has a relative minimum of - 23/2 at x = 1

Use the Second Derivative Test to locate any relative extrema, if they exist, for the function f(x)=2x^4-4x^2-5

f has a relative minimum of -7 at x = -1 f has a relative maximum of -5 at x = 0 f has a relative minimum of -7 at x = 1

Determine the absolute maximum of the function f(x) = -3x^2 + 1 - 5/x^2 on the interval (0, 10).

f has an absolute maximum at (4 square root of 5/3, -2 square root of 15 + 1) ≈ (1.14, -6.75)

Determine the absolute extrema of the function f(x) = x^2 - 10x - 8 on the interval [4, 8]

f has an absolute maximum of -33 at x = 5 f has an absolute minimum of -34 at x = 4

Determine the absolute extrema of the function f(x)=3 square root x+5 on the interval [-13,-4 ].

f has an absolute maximum of 1 at x = -4 f has an absolute minimum of -2 at x = -13

Determine the absolute extrema of the function f(x)= 1/x^2+3 on the interval [-2,2] .

f has an absolute maximum of 1/3 at x = 0 f has an absolute minimum of 1/7 at x = -2 and at x = 2

Determine the absolute extrema of the function f(x) = (3x - 1)^3 on the interval [-1, 2]

f has an absolute maximum of 125 at x = 2 f has an absolute minimum of -64 at x = -1

Use your graphing calculator to determine the absolute extrema of the function f(x)=x^3-4x^2+3x+2 on the interval [0,4] . Round answers to three decimal places, if necessary.

f has an absolute maximum of 14 at x = 4 f has an absolute minimum of -0.113 at x = 2.215

Determine the absolute extrema of the function f(x) =x/x-1 on the interval [2, 4]

f has an absolute maximum of 2 at x = 2 f has an absolute minimum of 4/3 at x = 4

Determine the absolute extrema of the function f(x) = 2x^3 +12x^2 - 7 on the interval [0, 4].

f has an absolute maximum of 313 at x = 4 f has an absolute minimum of -7 at x = 0

Determine the absolute extrema of the function f(x) = x^3 - 3x^2 - 9x on the interval [-4, 4]

f has an absolute maximum of 5 at x = -1 f has an absolute minimum of -76 at x = -4

Determine the absolute minimum of the function f(x) = 4x +6/x on the interval (0, 10).

f has an absolute minimum at (square root of 6/2, 4 square root of 6) ≈ (1.22, 9.80)

Determine the absolute minimum of the function f(x) = 4x - 3 + 7/x on the interval (0, 10).

f has an absolute minimum at (square root of 7/2, 4 square root of 7 - 3) ≈ (1.32,7.58)

The Computers and More store offers a service contract for customers purchasing a laptop. The service contract is paid for annually and the price increases each year that the buyer owns the laptop. Using the purchase price and the cost of the service contract, the function for the total cost for the laptop can be modeled by C(t) = 45t^2 + 70t + 2400 where t is the time in years and C(t) is the total cost of the laptop. i) The average cost function, denoted AC, is defined to be AC(t) =c(t)/t . Determine AC(t). ii) When is the average cost per year a minimum? Round to the nearest tenth. iii) What is the minimum average cost per year rounded to the nearest dollar?

i) AC(t) = 45t + 70 + 2400/t ii) 7.3 years iii) $727

The marketing research department for a manufacturing company has determined that the cost function for a particular product is given by C(x) = 0.02x^2 + 2x + 8, x > 0 i) Determine AC, the average cost function. ii) Determine the value of x that minimizes AC(x).

i) AC(x) = 0.02x + 2 +8/x ii) AC(x) is minimized at x ≈ 20 units

A company that manufactures video cards for computers determines that the weekly revenue can be modeled by R(x) = 27x - 0.01x^3 0 ≤ x ≤ 100 where x represents the number of video cards produced and sold each week and R(x) is the weekly revenue in dollars. i) Compute R'(x) and use it to determine intervals where R is increasing and where it is decreasing. ii) Determine intervals where the graph of R is concave up and where it is concave down.

i) R'(x) = 27 - 0.03x2; R is increasing on [0, 30) and decreasing on (30, 100]. ii) R is concave down on (0, 100]

Solve the problem. (picture) For the graph of f(x) above, i) Determine where f is increasing. ii) Determine where f' is increasing. iii) Locate any relative extrema. iv) Locate any inflection points.

i) f is increasing on (-5, -4) (-1, 2) (4, 5) ii) f' is increasing on (-5, -4) (-4, -3) (-2, 0) iii) f has relative maxima at (-4, 3), and (2, 3) f has relative minima at (-1, 0) and (4, 1) iv) f has inflection points at (-3, 2), (-2, 1) and (0, 1)