5.7 Solving Optimization Problems

• Identifying the objective function and constraints is crucial because they form the foundation for setting up and solving the problem
• Constraints are limitations or conditions that the variables in an optimization problem must satisfy
• What is the term for the mathematical expression that we aim to maximize or minimize in optimization problems?

  Objective function
• To express the objective function in terms of a single variable, you must use the constraint equation.

  True
• What is the feasible region in an optimization problem defined by constraints?

  Possible solutions satisfying constraints
• What type of constraint imposes a relationship between variables?

  Functional constraint
• Match the component with its description in the objective function P(x) = 10x - 2x:

  P(x) ↔️ Represents profit

  x ↔️ Number of units sold

  10x ↔️ Revenue from selling x units

  -2x ↔️ Production cost for x units
• The domain ensures the objective function produces valid outputs.

  True
• Critical points always represent the absolute maximum or minimum of the objective function.

  False
• The first derivative test is used to determine if a critical point is a maximum or a minimum
• A critical point occurs when A'(l) = 0
• An inequality constraint limits variables within a range
• What is the objective function in an optimization problem?

  Mathematical expression to optimize
• The domain of an objective function is the set of all possible values for the variables where the function is defined and meaningful.


  True
• Simplifying the objective function results in an equation with only one variable


  True
• Match the scenario with its domain:

  Area of a rectangle with perimeter 100 cm ↔️ 0 < l < 50

  Cost of packaging with volume 50 ↔️ l > 0, h > 0
• To find critical points, the first step is to take the derivative of the objective function.
• Steps to solve optimization problems

  1️⃣ Identify the objective function

  2️⃣ Identify the constraints

  3️⃣ Express the objective function in terms of a single variable

  4️⃣ Find critical points by taking the derivative and setting it to zero

  5️⃣ Determine maximum or minimum using the first or second derivative test

  6️⃣ Interpret the results
• The objective function represents the quantity to optimize in an optimization problem.

  True
• Match the constraint type with its description:

  Equality Constraint ↔️ Two expressions must be equal

  Inequality Constraint ↔️ Variables must be within a range

  Non-negativity Constraint ↔️ Variables must be non-negative

  Functional Constraint ↔️ Imposes a relationship between variables
• The objective function is written in terms of the problem's variables
• To express the objective function in terms of a single variable, first isolate a variable in the constraint
• An equality constraint requires two expressions to be equal
• Optimization problems involve finding the maximum or minimum value of the objective function.
• Steps to express an objective function in terms of a single variable:

  1️⃣ Isolate a variable in the constraint equation

  2️⃣ Substitute the isolated variable into the objective function

  3️⃣ Simplify the objective function
• Steps to find critical points of an objective function:

  1️⃣ Take the derivative of the function

  2️⃣ Set the derivative equal to zero

  3️⃣ Solve for x

  4️⃣ Consider undefined points
• The first derivative test is used to determine if a critical point is a maximum or a minimum.
• Checking the endpoints of the domain is necessary after finding critical points to determine absolute maxima or minima.

  True
• The maximum area of a rectangle with a perimeter of 100 cm is 625 square cm.

  True
• Match the constraint type with its description:

  Equality Constraint ↔️ Requires two expressions to be equal

  Inequality Constraint ↔️ Limits variables within a range

  Functional Constraint ↔️ Imposes a relationship between variables

  Non-negativity Constraint ↔️ Ensures variables are non-negative
• Maximizing profit can be represented by the objective function P(x) = 10x - x^2
• Steps to reduce variables in optimization problems

  1️⃣ Isolate a variable in the constraint equation

  2️⃣ Substitute the isolated variable into the objective function

  3️⃣ Simplify the objective function to have only one variable
• What is the domain of an objective function?

  All possible valid values
• Real-world limitations should be considered when determining the domain of an objective function


  True
• What type of rectangle maximizes the area when the perimeter is 100 cm?

  Square
• Why is it necessary to compare objective function values at critical points and endpoints?

  To find absolute maxima or minima
• Steps to find critical points of A(l) = 50l - l^2

  1️⃣ Take the derivative: A'(l) = 50 - 2l

  2️⃣ Set the derivative equal to zero: 50 - 2l = 0

  3️⃣ Solve for l: l = 25
• Checking the endpoints of the domain ensures the identification of absolute extrema.
• In the rectangle area example, at which length does the maximum area occur?

  25 cm
• Steps to state the final solution in an optimization problem

  1️⃣ Identify whether you found a maximum or minimum value

  2️⃣ Provide the maximum or minimum value of the objective function

  3️⃣ State the values of the variables at which the maximum or minimum occurs

  4️⃣ Include units of measure if appropriate