Optimization Involving a Single Variable

1. What is Optimization?

Optimization refers to the process of finding the maximum or minimum value of a function for a given set of constraints. In problems involving a single variable, calculus is used to locate these extreme values.

These problems often arise in geometry, economics, physics, and engineering.

2. Solving Strategy

1. Understand the problem and identify the quantity to be maximized or minimized.
2. Express this quantity as a function of a single variable.
3. Find the domain of the function based on physical or logical constraints.
4. Find the first derivative and determine critical points.
5. Use the first or second derivative test to classify the critical points.
6. Evaluate the function at critical points and endpoints to find the optimal value.

3. Example Problem

**Problem:** Find the dimensions of a rectangle with a perimeter of 100 units that has the maximum area.

1. Let length = \( x \), then width = \( (50 - x) \) because perimeter = 2(x + y).
2. Area \( A(x) = x(50 - x) = 50x - x^2 \)
3. Find the derivative: \( A'(x) = 50 - 2x \)
4. Set \( A'(x) = 0 \): \( 50 - 2x = 0 \Rightarrow x = 25 \)
5. Second derivative: \( A''(x) = -2 < 0 \), so it's a maximum.
6. So the maximum area occurs when both sides are 25 → a square.