Minimize

Minimize

To find the smallest possible value.

Example: Minimize f(x) = x² + 3. The smallest possible value of x² + 3 is 3, occurring when x = 0.

See also

Worked Example

Problem: A farmer has 40 meters of fencing and wants to enclose a rectangular pen against a barn wall (so only three sides need fencing). What dimensions minimize the amount of fencing used for a pen with an area of 200 square meters?

Set up: Let the width perpendicular to the barn be

x

and the length parallel to the barn be

y

. The three-sided perimeter is

P = 2x + y

, and the area constraint is

xy = 200

P = 2x + y, \quad xy = 200

Substitute: Solve the area equation for

y

and substitute into the perimeter expression to write

P

as a function of

x

alone.

P(x) = 2x + \frac{200}{x}

Find the critical point: Take the derivative and set it equal to zero to locate the minimum.

P'(x) = 2 - \frac{200}{x^2} = 0 \implies x^2 = 100 \implies x = 10

Solve for y: Substitute

x = 10

back into the area equation.

y = \frac{200}{10} = 20

Answer: The perimeter is minimized at

P = 2(10) + 20 = 40

meters, with width 10 m and length 20 m.

Why It Matters

Minimization appears throughout science, engineering, and everyday decision-making. Engineers minimize material costs, physicists minimize energy in a system, and businesses minimize expenses. Understanding how to minimize a quantity is one of the primary applications of calculus and algebra.

Common Mistakes

Mistake: Assuming every critical point is a minimum.

Correction: A critical point where the derivative equals zero could be a maximum or a saddle point. Use the second derivative test or check values on either side to confirm you have found a minimum.

Related Terms

Maximize — Opposite goal: finding the largest value
Minimum — The smallest value a function attains
Optimization — General process of minimizing or maximizing
Derivative — Key tool for finding where minima occur
Critical Point — Candidate location for a minimum