Optimization Problems: Maxima and Minima
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Questions and Answers

What is the value of x that maximizes the function A(x) = 2400x – 2x2?

  • 400
  • 600 (correct)
  • 800
  • 300

    • What is the value of the maximum of the function A(x) = 2400x – 2x2?

  • 960,000
  • 480,000
  • 720,000 (correct)
  • 360,000

    • What is the shape of the rectangular field that maximizes the function A(x) = 2400x – 2x2?

  • 1200 ft deep and 1200 ft wide
  • 600 ft deep and 600 ft wide
  • 1200 ft deep and 600 ft wide
  • 600 ft deep and 1200 ft wide (correct)

    • Why is the local maximum at x = 600 an absolute maximum?

    <p>The function is always concave downward</p> Signup and view all the answers

    What is the goal of the optimization problem in Example 2?

    <p>To minimize the cost of the metal to manufacture the can</p> Signup and view all the answers

    What is the shape of the sheet used to manufacture the cylindrical can?

    <p>Rectangular with dimensions 2πr and h</p> Signup and view all the answers

    What is the value of h when r = 500 / π?

    <p>1000</p> Signup and view all the answers

    What is the optimal radius of the can to minimize its cost?

    <p>3(500 / π) cm</p> Signup and view all the answers

    What is the condition for a critical number c to be an absolute maximum value of a function f?

    <p>f'(x) &gt; 0 for all x &lt; c and f'(x) &lt; 0 for all x &gt; c</p> Signup and view all the answers

    What is the purpose of implicit differentiation in optimization problems?

    <p>To find the critical numbers of a function</p> Signup and view all the answers

    What is the equation for the cost of the can in terms of radius and height?

    <p>A = 2πr^2 + 2πrh</p> Signup and view all the answers

    What is the name of the method used to determine the absolute minimum or maximum value of a function?

    <p>First Derivative Test</p> Signup and view all the answers

    What is the expression for the surface area of the cylinder in terms of the radius r?

    <p>A = 2πr^2 + 2πrh</p> Signup and view all the answers

    What is the value of h in terms of r, given that the volume of the cylinder is 1000 cm^3?

    <p>h = 1000 / (πr^2)</p> Signup and view all the answers

    What is the function A(r) that we want to minimize?

    <p>A(r) = 2πr^2 + r^2</p> Signup and view all the answers

    What is the critical number of the function A(r)?

    <p>r = 3√(500 / π)</p> Signup and view all the answers

    What can be concluded about the behavior of A(r) on the interval (0, ∞)?

    <p>A(r) is decreasing on (0, 3√(500 / π)) and increasing on (3√(500 / π), ∞)</p> Signup and view all the answers

    What is the nature of the critical number of the function A(r)?

    <p>Global minimum</p> Signup and view all the answers

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