Find the relative extrema of the function, if they exist.
1) f(x) = 2x2 + 20x + 53
2) f(x) = 6x/x2 + 1
3) f(x) = 3√(x + 1)
Determine where the given function is concave up and where it is concave down. Find the points of inflection.
4) f(x) = (x - 2)1/3 - 6
5) f(x) = x4 - 24x2
6) f(x) = -(2/3)x3 + 6x2 - x
Find the absolute maximum and absolute minimum values of the function, if they exist, over the indicated interval. When no interval is specified, use the real line (-∞,∞).
7) f(x) = x + (16/x); [- 6, - 1]
8) f(x) = - x2 + 11x - 30: [6, 5]
9) f(x) = (x - 3)3
Solve the problem.
10) A carpenter is building a rectangular room with a fixed perimeter of 140 ft. What are the dimensions of the largest room that can be built? What is its area?
11) A company wishes to manufacture a box with a volume of 36 cubic feet that is open on top and is twice as long as it is wide. Find the width of the box that can be produced using the minimum amount of material. Round to the nearest tenth, if necessary.
12) If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a certain city, where p(x) = 137 - x/18. How many candy bars must be sold to maximize revenue?
13) An appliance company determines that in order to sell x dishwashers, the price per dishwasher must be p = 720 - 0.3x. It also determines that the total cost of producing x dishwashers is given by C(x) = 6000 + 0.3x2. What price must be charged per dishwasher in order to maximize profit?
14) A bookstore has an annual demand for 63,000 copies of a best- selling book. It costs $0.40 to store one copy for one year, and it costs $65 to place an order. Find the optimum number of copies per order.
15) Given the total- revenue function R(x) = 5x, and the total- cost function C(x) = 0.01x2 + 2.2x + 80, if P(x) is the total profit function, find ΔP and P'(x) when x = 100 and Δx = 1.
16) A company estimates that the daily cost (in dollars) of producing x chocolate bars is given by C(x) = 595 + 0.04x + 0.0002x2. Currently, the company produces 260 chocolate bars per day. Use marginal cost to estimate the increase in the daily cost if one additional chocolate bar is produced per day.
Find dy/dx by implicit differentiation.
17) y√(x + 1) = 4
18) 2y - x + xy = 2
Find dy for the given values of x and dx.
19) y = 2x5 - 3x2 + x - 1; x = - 1, dx = 1/3
Solve the problem.
20) A spherical balloon is inflated with helium at a rate of 140π ft3/min. How fast is the balloon's radius increasing when the radius is 3 ft?
21) A ladder is slipping down a vertical wall. If the ladder is 20 ft long and the top of it is slipping at the constant rate of 4 ft/s, how fast is the bottom of the ladder moving along the ground when the bottom is 16 ft from the wall?