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Optimization Problems

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
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- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
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- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
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- Washer Method
- Decision Maths
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- 2 Dimensional Figures
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- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
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- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
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- Figures
- Fundamentals of Geometry
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- HL ASA and AAS
- Identity Map
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- Isometry
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- Law of Cosines
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- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
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- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
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- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
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- Projectiles
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- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
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- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
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- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
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- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
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- Distance from a Point to a Line
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- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
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- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
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- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
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- Points Lines and Planes
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- Powers and Roots
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- Problem-solving Models and Strategies
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- Proof
- Proof and Mathematical Induction
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- Segment of a Circle
- Sequences
- Sequences and Series
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- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
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- Simplifying Fractions
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- Simultaneous Equations
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- Small Angle Approximation
- Solving Linear Equations
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- Solving Radical Inequalities
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- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
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- Special Products
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- The Quadratic Formula and the Discriminant
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- Venn Diagrams

The primary idea in the business world is to maximize profit. However, it's not as simple as trying to sell as many products as possible. Other factors and costs go into a business, such as employee salaries, cost of production, cost of materials, and price of advertisement. Often, the answer to maximizing profit is **not** simply producing and selling as many products as possible.

Mathematical optimization can help find the answer that maximizes profit subject to the constraints of the real world. Optimization is one of the most interesting real-world applications of Calculus. This article will further define optimization, its other applications, and a method for solving simple optimization problems.

For a focus on business and economic-type optimization problems, see our article on Applications to Business and Economics

**Mathematical optimization **is the study of maximizing or minimizing a function subject to constraints, essentially finding the most effective and functional solution to a problem.

The constraints in optimization problems represent the limiting factors involved in the maximization/minimization problem. In our example of a business, the constraints would be the cost of labor, production, and advertisement. These constraints must be accounted for in our calculations as they can greatly influence the solution.

You've likely been learning and working through finding a function's extreme values (maximums and minimums). Optimization is a real-world application of finding and interpreting extreme values. Given an equation that models cost, we seek to find its minimum value, thus minimizing cost. Given an equation that models profits, we seek to find its maximum value, thus maximizing profit.

In addition to the business application we've discussed, optimization is crucial in various other fields. Optimization can be as simple as a traveler seeking to minimize transportation time. We can also apply optimization in medicine, engineering, financial markets, rational decision-making and game theory, and packaging shipments.

Optimization is also heavily discussed in computer science. Program optimization, space and time optimization, and software optimization are crucial in writing and developing efficient code and software.

Optimization problems can be quite complex, considering all the constraints involved. Converting real-world problems into mathematical models is one of the greatest challenges. As you progress through higher-level math classes, you'll deal with more complex optimization problems with more constraints to consider. In Calculus, we'll start with smaller-scale problems with fewer constraints. However, the baseline procedure is similar for all optimization problems.

Before we start working through optimization examples, we'll go through a general step-by-step method for working through these problems. Later on, we'll apply these steps as we work through real examples.

Optimization problems tend to pack loads of information into a short problem. The first step to working through an optimization problem is to read the problem carefully, gathering information on the known and unknown quantities and other conditions and constraints. It may be helpful to highlight certain values within the problem.

To better visualize the problem, it might be helpful to draw a diagram, including labels of known values provided in the problem.

Carefully declare variable names for values that are being maximized or minimized and other unknown quantities.

Use the known values and your declared variables to set up a function. You must set up your function in terms of these values and variables based on their relation to each other.

There are a couple of methods for finding absolute extrema in optimization problems.

If the domain of your function is a closed interval, the Closed Interval Method may be a good way to compute absolute extrema.

This method involves finding all critical values within the interval by setting and solving for . Each critical point, as well as the endpoints of the interval, should be plugged in to . The absolute extrema are largest value and smallest value of at the critical points.

The First Derivative Test for Absolute Extrema Values states that for a critical point of a function * *on an interval:

if for all and for all , then is the absolute maximum value of

if for all and for all , then is the absolute minimum value of

In other words, if the function goes from increasing to decreasing, it is a maximum. If the function goes from decreasing to increasing, it is a minimum.

Let's work through a common maximization problem.

You are tasked with enclosing a rectangular field with a fence. You are given 400 ft of fencing materials. However, there is a barn on one side of the field (thus, fencing is not required on one side of the rectangular field). What dimensions of the field will produce the largest area subject to the 400 ft of fencing materials?

We will solve this problem using the method outlined in the article.

Let's draw the important information out from the problem.

We need to fence **three sides** of a **rectangular **field such that the area of the field is **maximized**. However, we only have 400 ft of fencing material to use. Thus, the perimeter of the rectangle must be less than or equal to 400 ft.

Clearly, you don't have to be an artist to sketch a diagram of the problem!

Looking at the diagram above, we've already introduced some variables. We'll let the height of the rectangle be represented by . We'll let the width of the rectangle be represented by .

So, we can calculate area and perimeter as

The fencing problem wants us to maximize area *, *subject to the constraint that the perimeter must be greater or less than 400 ft. Intuitively, we know that we should use all 400 ft of fencing to maximize the area.

So, our problem becomes:

Since we seek to maximize the area, we must write the area in terms of the perimeter to achieve one single equation. In this example, we will write the area equation in terms of width, .

First, let's solve for the height, :

Now, plug into the area in terms of the width equation,

In this case, we solved for the variable *h* to write the area equation in terms of width. This is because solving for *h* does not yield a fractional answer, so it may be "easier" to work with for most students. It is entirely possible to solve for width and write the area equation in terms of height as well! Give it a try and see if you get the same answer!

Now that we have a single equation containing all of the information from the problem, we want to find the absolute maximum of . **We can define an interval for w so we can use the Closed Interval Method.**

For starters, we know that *w* cannot be smaller than 0. If we let , according to our perimeter equation, we have

This tells us that if , the maximum width possible is 200. So our closed interval for is .

To apply the Closed Interval Method:

First, find the extrema of by taking the derivative and setting it equal to 0.

Second, plug in the critical values and identify the largest area.

So, the largest value of occurs at where .

**We can confirm this using the First Derivative Test.**

Graphing ...

clearly only equals 0 at one point, . For all , is positive (above the *x*-axis). For all , is negative (below the *x*-axis). So, by the First Derivative Test, is the absolute maximum of .

Let's plug in to our perimeter equation to find out what *h* should be.

Therefore, to maximize the area enclosed by the fence subject to our material constraints, we should use a rectangle with a width of 100 ft and a height of 200 ft.

Now, let's try a minimization problem.

You are tasked with building a can that holds 1 liter of liquid. To maximize profit, you must build the can such that the material used to build it is minimized. What is the minimum surface area of the can required?

Again, we will solve this problem using the method outlined in the article.

Let's draw the important information out from the problem.

We need to build a can that holds 1 liter of liquid while minimizing the material used to build it. Essentially, this means we need to minimize the can's surface area.

With this diagram, we can better understand what the problem is asking us to do.

Looking at the diagram above, we've already introduced some variables. We'll let the radius of the cylindrical can be represented by . We'll let the height of the cylinder be represented by . So, the volume of the cylinder is and the surface area of the cylinder is .

The can problem wants us to minimize the surface area subject to the constraint that the can must hold at least 1 liter. Intuitively, we know that to minimize surface area, we should build a can that holds 1 liter of liquid. However, since we are looking for a length measurement for and , we should convert liters into cubic centimeters. Thus, we should build a can that holds 1,000 cm^{3} of liquid.

So, our problem becomes:

Since we seek to minimize the surface area, we must write the area in terms of the volume to achieve one single equation.

First, let's solve for *:*

Now, plug into the area equation:

Now that we have a single equation containing all the information from the problem, we want to find the absolute minimum of .

We know that . However, we do not have an upper bound for .

First, we'll find the extrema of by taking the derivative and setting it equal to 0.

Graphing the derivative:

We can see* * at one point. We can confirm that the point is an absolute minimum for by applying the **First Derivative Test**. Looking at the graph, For all , is negative (below the *x*-axis). For all , is positive (above the *x*-axis). So, by the First Derivative Test, is the absolute maximum of .

Let's plug in to our volume equation to find out with should be.

So, to build a can that holds at least 1 liter, the minimum surface area required is

**Mathematical optimization**is the study of maximizing or minimizing a function subject to constraints, essentially finding the most effective and functional solution to a problem- Optimization is a real-world application of finding and interpreting extreme values

- Solving optimization problems can seem daunting at first, but following a step-by-step procedure helps:
- Step 1: Fully understand the problem
- Step 2: Draw a diagram
- Step 3: Introduce necessary variables
- Step 4: Set up the problem by finding relationships within the problem
- Step 5: Find the absolute extrema

- To find the absolute extrema, use either the Closed Interval Method or the First Derivative Test

**maximizing **or **minimizing **certain quantities. To determine if a problem is an optimization problem, carefully read the problem and look for language that suggests maximizing or minimizing.

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