Optimization Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Optimization.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The process of using derivatives to systematically find maximum or minimum values of a function over a domain.

Find where the function hits its peaks (maxima) and valleys (minima) by finding where the slope is zero.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: At a local maximum or minimum, the derivative equals zero (or is undefined) — these are called critical points.

Common stuck point: Check endpoints too—max/min might be at boundaries, not where derivative = 0.

Sense of Study hint: Draw a labeled diagram of the situation, write one equation for what you optimize and one for the constraint.

Worked Examples

Example 1

easy
Find the local maximum and minimum values of f(x) = x^3 - 3x + 2.

Solution

  1. 1
    Find f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x-1)(x+1).
  2. 2
    Set f'(x) = 0: critical points at x = 1 and x = -1.
  3. 3
    Find f''(x) = 6x. At x = -1: f''(-1) = -6 < 0, so local maximum.
  4. 4
    At x = 1: f''(1) = 6 > 0, so local minimum.
  5. 5
    Evaluate: f(-1) = -1 + 3 + 2 = 4 (local max); f(1) = 1 - 3 + 2 = 0 (local min).

Answer

Local maximum: f(-1) = 4; local minimum: f(1) = 0
The second derivative test classifies critical points: negative second derivative means concave down (local max), positive means concave up (local min). Always evaluate the function at the critical point to get the actual max/min value.

Example 2

hard
A farmer has 200 m of fencing to enclose a rectangular field. What dimensions maximize the enclosed area?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the critical points of f(x) = 2x^3 - 9x^2 + 12x - 4 and classify them.

Example 2

medium
Find the absolute maximum and minimum of f(x) = x^3 - 6x^2 + 9x + 1 on [0, 4].
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Related Concepts

Derivative

Background Knowledge

These ideas may be useful before you work through the harder examples.

derivative