Chapter 6 · Test 1
One-Variable Optimization for Economics
First- and second-order conditions and optimization on intervals.
What you will learn
- Find an unconstrained maximum
- Optimize on a closed interval
- Use the second-order condition
Worked example
Find an unconstrained maximum: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Find an unconstrained maximum
Profit is π(q) = −2q² + 40q − 50. Find the profit-maximizing quantity q*.
Interior first-order condition: π′(q*) = 0
In plain English: At an interior profit optimum, a very small change in quantity has no first-order effect on profit, so marginal profit is zero.
Bold blue terms in this symbol guide include definitions. Hover, focus, or tap a term. Each row also shows how to read the notation aloud.
- q
- output quantityHow to read: output quantity
- π(q)
- Profit: Total revenue minus total cost. Economic profit subtracts both explicit costs and implicit opportunity costs. at output qHow to read: profit as a function of output
- q*
- the quantity that maximizes Profit: Total revenue minus total cost. Economic profit subtracts both explicit costs and implicit opportunity costs.How to read: profit-maximizing quantity
Solution
- Differentiate: π′(q) = −4q + 40.
- Set the first derivative to zero: −4q + 40 = 0.
- Solve: −4q = −40, so q* = 10.
- Verify: π″(q) = −4 < 0, so the stationary point is a strict maximum.
Visual check
Profit reaches its maximum
The vertex of the downward-opening parabola is the profit-maximizing quantity.
- Objective
How to read this graph
- Horizontal axis: Quantity, q
- Vertical axis: Profit, π(q)
- Curves and lines: Objective
- Marked points: Open the exact-coordinate table below to read their values.
What the graph shows: Maximum profit at Quantity, q = 10, where the objective equals 150.
Read the marked points as data
| Feature | Quantity, q | Profit, π(q) |
|---|---|---|
| Maximum profit | 10 | 150 |
Logic and solving tips
- A zero first derivative locates a candidate; negative curvature verifies a local maximum.
- Solve the first-order condition for every candidate, then evaluate the second derivative before classifying the result.
Where economists use it
A firm optimizes a profit function to choose the output, price, or advertising level that produces the highest attainable profit.
Key idea: A zero first derivative locates a candidate; negative curvature verifies a local maximum.
Applied case study
Choosing a generator's profit-maximizing output
Scenario
Energy economists compare marginal revenue with marginal cost when modeling a price-taking generator. FERC explains that RTO/ISO markets dispatch resources in cost order and that the locational marginal price reflects the cost of the next unit of energy. Here one generator faces a teaching price of $70 per MWh.
Problem
If C(q) = 10q + q², what hourly output q maximizes profit?
π(q) = 70q - C(q) = 60q - q² π′(q) = 60 - 2q π″(q) = -2
In plain English: Profit is written as revenue minus cost; its first derivative locates stationary quantities and its second derivative shows whether profit bends upward or downward there.
Worked solution
- Write profit as revenue minus cost: π(q) = 70q - (10q + q²).
- Set marginal profit to zero: 60 - 2q = 0.
- Solve to obtain q* = 30 MWh.
- Because π″(q) = -2 < 0, q* is a strict maximum; π(30) = $900.
Economic interpretation: Under the simplified price and cost curve, 30 MWh is the generator's profit-maximizing hourly output.
Source: U.S. Federal Energy Regulatory Commission, Energy Markets FERC supplies the market setting, dispatch rule, and marginal-price interpretation. The price and quadratic cost curve are teaching assumptions, not data for a named plant; startup costs, congestion, losses, and strategic bidding are omitted.
Reasoning habits that help
- A zero first derivative locates a candidate; negative curvature verifies a local maximum.
- On a closed interval, endpoints are candidates even when they do not satisfy the interior first-order condition.
- The first derivative finds candidates; the second derivative classifies them through curvature.
Where economists use these methods
- A firm optimizes a profit function to choose the output, price, or advertising level that produces the highest attainable profit.
- A factory searches both interior choices and endpoints when output is limited by capacity, a budget, regulation, or nonnegative production.
- After finding a zero marginal condition, an economist checks curvature to verify that it is a profit maximum or cost minimum rather than the wrong turning point.
Chapter vocabulary
Study 27 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Optimization
The process of finding the feasible choice that gives the greatest or least value of an objective function.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed page 195Objective function
The function an optimization problem seeks to maximize or minimize, such as profit, utility, cost, or loss.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed page 195Unconstrained extremum
A maximum or minimum found without restrictions beyond the function's domain.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed pages 195-196Local maximum
A point whose objective value is at least as high as all sufficiently nearby feasible values.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed pages 196-197Local minimum
A point whose objective value is no greater than all sufficiently nearby feasible values.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed pages 196-197Global maximum
A feasible point whose objective value is at least as high as the value at every other feasible point.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed pages 196-197Global minimum
A feasible point whose objective value is no greater than the value at every other feasible point.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed pages 196-197Extreme value
A function value attained at a local or global maximum or minimum.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed page 197Interior solution
An optimizing choice that lies inside the feasible domain rather than on its boundary.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed pages 197-198Stationary valuef'(x*) = 0
An input value x* at which the first derivative equals zero. Its graph point can be a maximum, a minimum, or a stationary inflection point.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed page 200Point of inflection
A point where the function changes between convex and concave curvature. Its first derivative need not be zero.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed page 199First-order conditionf'(x*) = 0
A condition involving first derivatives that an interior optimum candidate must satisfy.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed page 197Necessary condition
A requirement every solution of the stated type must satisfy, even though satisfying it may not guarantee a solution.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed pages 200-201Sufficient condition
A requirement that guarantees the stated conclusion, even though the conclusion may also hold in other cases.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed pages 200-201Profit maximizationmax π(q) = R(q) - C(q)
Choosing output, price, or inputs to make total revenue minus total economic cost as large as possible.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed pages 203, 205Marginal revenue equals marginal costMR(q*) = MC(q*)
The first-order condition for an interior output choice when profit is differentiable.
Textbook reference: Chapter 6, Section 6.1 Necessary Conditions for Unconstrained Maxima and Minima, printed pages 203, 205Second-order condition
A condition using the second derivative to distinguish the curvature associated with a maximum or minimum candidate.
Textbook reference: Chapter 6, Section 6.2 Second-Order Conditions, printed page 211Second derivative test
At a stationary point, a negative second derivative is sufficient for a strict local maximum and a positive one for a strict local minimum.
Textbook reference: Chapter 6, Section 6.2 Second-Order Conditions, printed pages 211-212nth derivative test
When lower derivatives vanish at a stationary point, the first nonzero higher derivative classifies the point by its order and sign.
Textbook reference: Chapter 6, Section 6.2 Second-Order Conditions, printed page 217Flat stationary point
A stationary point at which the second derivative also vanishes, requiring higher-order or other analysis.
Textbook reference: Chapter 6, Section 6.2 Second-Order Conditions, printed pages 213-214Stationary point of inflection
A stationary point where curvature changes sign but the function has neither a local maximum nor a local minimum.
Textbook reference: Chapter 6, Section 6.2 Second-Order Conditions, printed pages 214-215Constrained extremum
A maximum or minimum found while the choice must satisfy one or more restrictions.
Textbook reference: Chapter 6, Section 6.3 Optimization over an Interval, printed page 220Endpoint solution
An optimum attained at an endpoint of the permitted interval rather than at an interior stationary point.
Textbook reference: Chapter 6, Section 6.3 Optimization over an Interval, printed pages 220-221Optimization over a closed interval
Comparing objective values at all interior stationary points and both endpoints of the interval.
Textbook reference: Chapter 6, Section 6.3 Optimization over an Interval, printed pages 220-221Binding constraint
A restriction satisfied as an equality at the optimum that prevents the unconstrained preferred choice.
Textbook reference: Chapter 6, Section 6.3 Optimization over an Interval, printed page 228Nonbinding constraint
A restriction that does not limit the optimum, so relaxing it slightly would not improve the objective.
Textbook reference: Chapter 6, Section 6.3 Optimization over an Interval, printed pages 227-228Shadow price
The marginal improvement in the optimized objective from a small relaxation of a binding constraint, under regularity conditions.
Example: A capacity constraint's shadow price measures the extra profit available from one more unit of capacity.
Textbook reference: Chapter 6, Section 6.3 Optimization over an Interval, printed page 227End-of-chapter problems
Practice all 10 Chapter Review Exercises
The index covers every numbered Review Exercise in this chapter of Mathematics for Economics, 3rd edition. Each reference opens the textbook exercise itself, including every listed subpart, with MFE hints, a symbol guide, a worked explanation, progress tracking, and similar-problem practice.
Practice end-of-chapter problems