01 Key Concepts
Objective Function
The quantity you're trying to maximize or minimize, such as profit, cost, or distance.
Constraints
Limits or restrictions on the variables involved, such as a fixed budget or limited time.
Unconstrained Optimization
Finding a maximum or minimum using calculus (setting the derivative to zero), with no restrictions on the variable's range.
Constrained Optimization
Finding the best outcome while satisfying one or more constraints, often visualized as finding the best point within a feasible region.
Linear Programming
A specific type of constrained optimization where both the objective function and constraints are linear; solutions occur at corners of the feasible region.
02 Solved Examples
Example 1 A company's profit is P(x) = -2x^2 + 80x - 300, where x is units sold. Find the number of units that maximizes profit.
- Take the derivative: P'(x) = -4x + 80.
- Set P'(x)=0: -4x+80=0, so x=20.
Answer: 20 units maximizes profit
Example 2 Using the previous example, find the maximum profit.
- Substitute x=20 into P(x): P(20) = -2(20)^2 + 80(20) - 300.
- = -800 + 1600 - 300.
Answer: $500
Example 3 A factory has a budget constraint: it can spend at most $1000 on materials at $50 each. What is the constraint's maximum quantity?
- Divide the budget by the cost per unit: 1000/50.
Answer: 20 units maximum, given the budget constraint
03 Practice Questions
1What is the term for the quantity being maximized or minimized?
The objective function
2What are 'constraints' in an optimization problem?
Limits or restrictions on the variables
3For P(x)=-x^2+40x, find the x that maximizes P (Hint: P'(x)=-2x+40).
x=20
4What type of optimization has both a linear objective function and linear constraints?
Linear programming
5Where do solutions to linear programming problems typically occur?
At a corner (vertex) of the feasible region
📄 Optimization — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.