Business and Economics Optimization Calculator

Math Calculus • Applications of Derivatives

Written by STEM Calculators Team Published June 15, 2026 Updated June 15, 2026

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Theory

Marginal analysis model

Business optimization usually compares cost, revenue, and profit. The quantity \(x\) represents the number of units produced or sold.

Step 1. Define cost, revenue, and profit.

\[ \begin{aligned} C(x) &= \text{cost function} \\ R(x) &= \text{revenue function} \\ P(x) &= R(x)-C(x) \end{aligned} \]

Step 2. Use marginal quantities.

Marginal quantities are derivatives. They measure approximate change per extra unit.

\[ \begin{aligned} MC(x) &= C'(x) \\ MR(x) &= R'(x) \\ MP(x) &= P'(x) \end{aligned} \]

Step 3. Relate marginal profit to marginal revenue and marginal cost.

\[ \begin{aligned} P(x) &= R(x)-C(x) \\ P'(x) &= R'(x)-C'(x) \\ MP(x) &= MR(x)-MC(x) \end{aligned} \]

Therefore, at an interior profit maximum, the usual condition is:

\[ \begin{aligned} MP(x) &= 0 \\ MR(x)-MC(x) &= 0 \\ MR(x) &= MC(x) \end{aligned} \]

Important: \(MR=MC\) identifies an interior candidate. Endpoints must still be checked on a closed interval.

Closed-interval optimization method

If production is restricted to a closed interval \([a,b]\), the calculator uses the same closed-interval method used in calculus optimization.

Step 1. Choose the objective function.

The objective may be profit, revenue, cost, or average cost.

\[ \begin{aligned} F(x) &= \begin{cases} P(x), & \text{profit maximization} \\ R(x), & \text{revenue maximization} \\ C(x), & \text{cost minimization} \\ AC(x), & \text{average cost minimization} \end{cases} \end{aligned} \]

Step 2. Find interior critical points.

\[ \begin{aligned} F'(c) &= 0 \\ a &< c < b \end{aligned} \]

Step 3. Build the candidate set.

\[ \begin{aligned} S &= \{a,b,c_1,c_2,\ldots,c_n\} \end{aligned} \]

Step 4. Evaluate and compare.

\[ \begin{aligned} \text{maximum objective value} &= \max\{F(x):x\in S\} \\ \text{minimum objective value} &= \min\{F(x):x\in S\} \end{aligned} \]

Worked profit-maximization example

Suppose \(C(x)=x^2+10x\) and \(R(x)=50x-x^2\). Find the profit maximum.

Step 1. Build the profit function.

\[ \begin{aligned} P(x) &= R(x)-C(x) \\ &= (50x-x^2)-(x^2+10x) \\ &= 40x-2x^2 \end{aligned} \]

Step 2. Differentiate profit.

\[ \begin{aligned} P'(x) &= 40-4x \end{aligned} \]

Step 3. Set marginal profit equal to zero.

\[ \begin{aligned} 40-4x &= 0 \\ 40 &= 4x \\ x &= 10 \end{aligned} \]

Step 4. Evaluate profit at the critical point.

\[ \begin{aligned} P(10) &= 40(10)-2(10)^2 \\ &= 400-200 \\ &= 200 \end{aligned} \]

The profit is maximized at \(x=10\), with maximum profit \(200\).

Elasticity and average cost

If revenue is \(R(x)\), then the average price per unit can be estimated by \(p(x)=R(x)/x\), when \(x>0\).

\[ \begin{aligned} p(x) &= \frac{R(x)}{x} \end{aligned} \]

One common elasticity model is:

\[ \begin{aligned} E(x) &= -\frac{p(x)}{x\,p'(x)} \end{aligned} \]

Average cost is:

\[ \begin{aligned} AC(x) &= \frac{C(x)}{x} \end{aligned} \]

Minimizing average cost can help identify the production level with the lowest cost per unit.

Common mistakes

Forgetting endpoints: closed-interval optimization must include both endpoints.
Confusing revenue and profit: maximum revenue is not always maximum profit.
Using \(MR=MC\) without checking: \(MR=MC\) gives an interior candidate, not automatically the final answer.
Ignoring domain restrictions: average cost and price formulas usually require \(x>0\).
Comparing \(x\)-values instead of objective values: the best result is decided by the value of the objective function.

Frequently Asked Questions

How do you maximize profit?

Build the profit function P(x)=R(x)-C(x), find critical points where P'(x)=0, evaluate endpoints and critical points, and choose the largest profit value.

What does MR=MC mean?

MR=MC means marginal revenue equals marginal cost. At an interior profit maximum, this is the usual marginal condition.

What is marginal cost?

Marginal cost is C'(x), the approximate extra cost of producing one more unit.

What is marginal revenue?

Marginal revenue is R'(x), the approximate extra revenue from selling one more unit.

What is marginal profit?

Marginal profit is P'(x)=R'(x)-C'(x). It measures how profit changes when quantity increases.

What is average cost?

Average cost is AC(x)=C(x)/x, the cost per unit produced.

Can the calculator estimate elasticity?

Yes. If price can be approximated by p(x)=R(x)/x, then elasticity is estimated as E=-p/(x p'), when defined.

For C(x)=x^2+10x and R(x)=50x-x^2, where is profit maximized?

Profit is P(x)=40x-2x^2, so P'(x)=40-4x. The interior maximum occurs at x=10, with profit P(10)=200.

How can I get a profit maximum at x=20?

One example is C(x)=x^2+10x and R(x)=90x-x^2. Then P(x)=80x-2x^2 and P'(x)=80-4x, so the maximum occurs at x=20.

Business and Economics Optimization Calculator

Cost, revenue, and optimization interval

Graph and output settings

Quick examples

Business optimization graph

Step-by-step marginal optimization

Rate this calculator

Frequently Asked Questions

Cost, revenue, and optimization interval

Graph and output settings

Quick examples

Business optimization graph

Step-by-step marginal optimization

Rate this calculator

Frequently Asked Questions

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