Mathematics 133 -- Intensive Calculus

Problem Set 8 Answers

November 11, 2000

• A) For each of the following functions,
  • find all critical points,
  • determine the intervals on which f(x) is increasing or decreasing,
  • identify each critical point as a local maximum, local minimum, or neither,
  • determine the intervals on which y = f(x) is concave up or down,
  • determine all inflection points
  
  
  1. f(x) = 2x³ - 3x² - 12x: critical points, -1,2, increasing on (-infinity, -1) and (2,infinity), decreasing on (-1,2), -1 is a local maximum, 2 is a local minimum, concave down on (-infinity,1/2), concave up on (1/2,infinity), 1/2 is the only inflection point
  2. f(x) = x sqrt(x² + 1): increasing for all x, no critical points, concave down for all x < 0, up for all x > 0, inflection point at x = 0
  3. f(x) = x - 2 sin(x) for 0 < x < 3Pi: critical points: x = Pi/3, 5Pi/3, 7Pi/3; increasing on (Pi/3,5Pi/3), (7Pi/3,3Pi), decreasing on (0,Pi/3),(5Pi/3,7Pi/3), Pi/3, 7Pi/3 are local minima, 5Pi/3 is a local maximum, inflection points x = Pi, 2Pi.
• B) Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible: Answer: square with side 25 m.
• C) Find the point on the line y = 4x+7 that is closest to the origin. Answer: (-28/17, 7/17).
• D) A piece of wire 10 m long is cut into two pieces. The first is bent to form a square, and the second to form an equilateral triangle. What are the maximum and minimum total areas that can be enclosed. Answer: Maximum when all the wire is used for the square, minimum when 4.35 m of the wire is used for the square.
• E) A cylindrical can without a top is made to contain V cubic inches of liquid. Find the dimensions that will minimize the cost of the can, assuming the metal costs a constant amount per square inch. Answer: height = radius = cuberoot(V/Pi).
• F) A conical drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges. Find the maximum capacity of such a cup. Answer: 2Pi R³/(9 sqrt(3))