Maximize

Maximize

To find the largest possible value.

Example: Maximize f(x) = 4 − x². The largest possible value of 4 − x² is 4, occurring when x = 0.

See also

Worked Example

Problem: A farmer has 40 meters of fencing and wants to enclose a rectangular garden along a wall (so only three sides need fencing). What dimensions maximize the area of the garden?

Step 1: Let the width be

x

meters and the length be

y

meters. Since one long side is against the wall, the fencing constraint is:

2x + y = 40

Step 2: Solve for

y

and write the area

A

as a function of

x

A = x \cdot y = x(40 - 2x) = 40x - 2x^2

Step 3: This is a downward-opening parabola. Its maximum occurs at the vertex, where

x = -\frac{b}{2a}

x = -\frac{40}{2(-2)} = 10

Step 4: Find

y

and the maximum area:

y = 40 - 2(10) = 20, \quad A = 10 \times 20 = 200

Answer: The area is maximized at 200 square meters, with width 10 m and length 20 m.

Why It Matters

Maximizing appears throughout real-world decisions: businesses maximize profit, engineers maximize efficiency, and designers maximize strength within material limits. In algebra and calculus, learning to maximize a function teaches you how to analyze its behavior and locate its highest point, which is a foundational skill in optimization.

Common Mistakes

Mistake: Ignoring constraints and finding a value that is not actually achievable.

Correction: Always check that your answer satisfies all given constraints (such as positive dimensions or a budget limit). The maximum of a function without constraints may differ from the maximum within a restricted domain.

Related Terms

Minimize — The opposite goal: finding the smallest value
Maximum of a Function — The highest point on a function's graph
Optimization — The broader field of maximizing or minimizing
Vertex — Location of a parabola's maximum or minimum