Optimization Challenges for Students

What is the maximum value of the function f(x) = -2x^2 + 4x + 1 within the interval [0, 3]?

• 1

• 4

• 2

• 3

Minimize the cost function C(x) = 5x + 3 subject to the constraint x ≥ 2. What is the minimum cost?

• 10

• 20

• 13

• 15

Given the linear programming problem, maximize z = 3x + 4y subject to the constraints x + 2y ≤ 8 and x ≥ 0, y ≥ 0. What is the maximum value of z?

• 16

• 10

• 20

• 12

A farmer wants to fence a rectangular area of 200 square meters. What dimensions will maximize the area if the length is twice the width?

• Length: 25 meters, Width: 12.5 meters

• Length: 15 meters, Width: 5 meters

• Length: 20 meters, Width: 10 meters

• Length: 30 meters, Width: 15 meters

Find the minimum value of the function g(x) = x^2 - 6x + 10. What is the minimum value and at which x does it occur?

• Minimum value is 1 at x = 3.

• Minimum value is 0 at x = 1.

• Minimum value is 4 at x = 2.

• Minimum value is 2 at x = 4.

Maximize the profit function P(x) = 50x - 2x^2 under the constraint that x ≤ 20. What is the maximum profit?

• 312.5

• 400

• 300

• 250

A company produces two products A and B. The profit from A is $5 and from B is $7. If the production constraints are 3A + 2B ≤ 12, what is the maximum profit?

• 35

• 50

• 42

• 28

Determine the maximum area of a triangle with a base of 10 meters and a height that can vary between 0 and 8 meters. What is the maximum area?

• 40 square meters

• 20 square meters

• 30 square meters

• 50 square meters

Minimize the function h(x) = 4x^2 - 8x + 5. What is the minimum value and the corresponding x?

• Minimum value: 0, corresponding x: 0

• Minimum value: 2, corresponding x: 2

• Minimum value: 3, corresponding x: -1

• Minimum value: 1, corresponding x: 1

A delivery truck can carry a maximum of 1000 kg. If each package weighs 5 kg, how many packages can be maximized?

• 150

• 300

• 200

• 250