BitWithBite
BitWithBite / Calculus / Optimization
📘 Lesson 8 of 20 · Calculus
Differential Calculus

🏆 Optimization

Optimization uses derivatives to find the best possible outcome — the maximum profit, minimum cost, or largest area — subject to given constraints.

Course progress: 40%

01 Key Concepts

General Strategy

1) Write a formula for the quantity to optimize. 2) Use a constraint to reduce it to one variable. 3) Differentiate, set equal to zero, and solve. 4) Confirm it's a max or min.

First Derivative Test

If f' changes from positive to negative at a critical point, it's a local max; from negative to positive, a local min.

Second Derivative Test

If f''(c) > 0 at a critical point c, it's a local minimum. If f''(c) < 0, it's a local maximum.

Constraints

A limiting condition (like fixed perimeter or fixed cost) used to express one variable in terms of another, reducing the problem to a single variable.

Endpoints Matter

On a closed interval, always also check the endpoints — the true maximum or minimum might occur there instead of at a critical point.

02 Solved Examples

Example 1 A farmer has 40m of fencing to enclose a rectangular field. Find the dimensions that maximize area.
  1. Let width = x, length = y. Constraint: 2x + 2y = 40, so y = 20 - x.
  2. Area A = x*y = x(20-x) = 20x - x^2.
  3. Differentiate: A' = 20 - 2x. Set to 0: x = 10.
  4. Then y = 20 - 10 = 10.
Answer: 10m by 10m (a square), giving maximum area 100 sq m
Example 2 Find two numbers whose sum is 20 and whose product is maximum.
  1. Let the numbers be x and 20-x. Product P = x(20-x) = 20x - x^2.
  2. Differentiate: P' = 20 - 2x. Set to 0: x = 10.
  3. The other number is 20 - 10 = 10.
Answer: Both numbers are 10, maximum product = 100
Example 3 Minimize the cost function C(x) = x^2 - 20x + 150.
  1. Differentiate: C'(x) = 2x - 20. Set to 0: x = 10.
  2. Check second derivative: C''(x) = 2 > 0, confirming a minimum.
  3. Minimum cost = C(10) = 100 - 200 + 150.
Answer: Minimum cost = $50 at x = 10

03 Practice Questions

1A rectangle has perimeter 60. Find dimensions that maximize area.
15 by 15
2Find two numbers with sum 30 whose product is maximum.
15 and 15
3Minimize C(x) = x^2 - 10x + 40.
Minimum at x=5, C=15
4What test uses f''(c) to classify a critical point?
The second derivative test
5When maximizing area with fixed perimeter, what shape is usually optimal for a rectangle?
A square

📄 Optimization — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet