Optimization Problems: Maxima and Minima

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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?

The function is always concave downward Signup and view all the answers

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

To minimize the cost of the metal to manufacture the can Signup and view all the answers

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

Rectangular with dimensions 2πr and h Signup and view all the answers

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

1000 Signup and view all the answers

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

3(500 / π) cm 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?

f'(x) > 0 for all x < c and f'(x) < 0 for all x > c Signup and view all the answers

What is the purpose of implicit differentiation in optimization problems?

To find the critical numbers of a function Signup and view all the answers

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

A = 2πr^2 + 2πrh 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?

First Derivative Test Signup and view all the answers

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

A = 2πr^2 + 2πrh 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?

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

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

A(r) = 2πr^2 + r^2 Signup and view all the answers

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

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

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

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

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

Global minimum Signup and view all the answers