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Quiz: Optimization

Test your understanding of optimization with these review questions.

1. The function you want to maximize or minimize in an optimization problem is called the:

  1. Constraint equation
  2. Objective function
  3. Critical function
  4. Primary variable
Show Answer

The correct answer is B. The objective function is what you're trying to optimize (maximize or minimize). The constraint equation relates the variables and allows you to reduce to one variable.

Concept Tested: Objective Function

2. The first step in solving an optimization problem is to:

  1. Take the derivative
  2. Set the derivative equal to zero
  3. Identify what to optimize and draw a diagram
  4. Find the second derivative
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The correct answer is C. Always start by understanding what you're optimizing, identifying variables, and drawing a diagram if applicable. Then write the objective function and constraints.

Concept Tested: Setting Up Optimization

3. A constraint equation is used to:

  1. Find the derivative
  2. Express the objective function in terms of one variable
  3. Determine if a function is continuous
  4. Find the second derivative
Show Answer

The correct answer is B. The constraint equation relates the variables. Use it to eliminate one variable, writing the objective function in terms of a single variable so you can take its derivative.

Concept Tested: Constraint Equation

4. What is the maximum area of a rectangle with perimeter 100 feet?

  1. 500 sq ft
  2. 625 sq ft
  3. 1000 sq ft
  4. 2500 sq ft
Show Answer

The correct answer is B. Constraint: 2L + 2W = 100, so W = 50 − L. Objective: A = L·W = L(50 − L) = 50L − L². A' = 50 − 2L = 0 gives L = 25, W = 25. A = 625 sq ft (a square).

Concept Tested: Maximizing Area

5. After finding critical points in an optimization problem, you should:

  1. Stop—the critical point is always the answer
  2. Verify it gives a maximum or minimum (not just a critical point)
  3. Ignore the constraint
  4. Find more critical points
Show Answer

The correct answer is B. A critical point might be a max, min, or neither. Use the Second Derivative Test, First Derivative Test, or compare endpoint values to verify you found the desired extremum.

Concept Tested: Verifying Maximum

6. In a closed-box problem, you want to minimize surface area given a fixed volume. The constraint is:

  1. Surface area formula
  2. Volume formula set equal to a constant
  3. Perimeter formula
  4. Cost formula
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The correct answer is B. When volume is fixed, V = lwh = constant is the constraint. Surface area S = 2lw + 2lh + 2wh is the objective function to minimize.

Concept Tested: Box Problem

7. A farmer has 200 meters of fencing to enclose a rectangular area against a river (no fence needed on river side). What dimensions maximize area?

  1. 50m × 100m
  2. 100m × 50m
  3. 50m × 50m
  4. 66.67m × 66.67m
Show Answer

The correct answer is A. Constraint: 2W + L = 200 (three sides). Objective: A = L·W = (200 − 2W)W = 200W − 2W². A' = 200 − 4W = 0 gives W = 50, L = 100. Area = 5000 m².

Concept Tested: Fencing Problem

8. The practical domain in an optimization problem refers to:

  1. All real numbers
  2. Values that make physical sense in the problem context
  3. Only positive numbers
  4. The domain of the derivative
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The correct answer is B. The practical domain includes only values that make sense in context (positive lengths, non-negative quantities, etc.). This may restrict where critical points are valid.

Concept Tested: Practical Domain

9. To minimize the distance from a point to a curve, you can instead minimize:

  1. The x-coordinate
  2. The square of the distance
  3. The y-coordinate
  4. The slope of the line
Show Answer

The correct answer is B. Minimizing d² gives the same location as minimizing d (since d ≥ 0), but d² is easier to differentiate because it avoids the square root.

Concept Tested: Minimizing Distance

10. A company's profit is P(x) = R(x) − C(x). Maximum profit occurs when:

  1. R(x) = C(x)
  2. R'(x) = C'(x) (marginal revenue = marginal cost)
  3. R(x) is maximum
  4. C(x) is minimum
Show Answer

The correct answer is B. P'(x) = R'(x) − C'(x) = 0 means R'(x) = C'(x). Maximum profit occurs when marginal revenue equals marginal cost—a fundamental result in economics.

Concept Tested: Maximum Profit