Optimization

You’ll be able to

Translate a word problem into an objective function of one variable
Use critical points and endpoint analysis to find absolute extrema
Justify that a critical point is a maximum or minimum

Optimization finds a best value

An optimization problem asks for the largest or smallest possible value of some quantity — maximum area, minimum cost, shortest distance — subject to a constraint. The strategy is to build an objective function for the quantity being optimized, use the constraint to reduce it to a single variable, then find its absolute extremum on the relevant domain. Calculus enters because extrema occur where the derivative is zero or undefined, or at the endpoints of the domain.

Critical points and the candidates test

A critical point is where f′(x) = 0 or f′ is undefined; these are the only interior candidates for a maximum or minimum. To find an absolute extremum on a closed interval, evaluate f at every critical point and at both endpoints — the largest output is the absolute max, the smallest the absolute min. This "candidates test" is bulletproof because a continuous function on a closed interval is guaranteed to attain both extremes (Extreme Value Theorem).

First-derivative test for a maximum

If f′ changes + → − at c, then f has a local maximum at c.

A sign change from positive to negative means the function rises then falls — a peak. The reverse (− → +) is a local minimum.

Worked example

A farmer has 200 m of fencing to enclose a rectangular field against a straight river (no fence on the river side). What dimensions maximize the area?

1.Let x be each of the two sides perpendicular to the river and y the side parallel to it. Fencing used: 2x + y = 200, so y = 200 − 2x.
2.Objective: area A = x·y = x(200 − 2x) = 200x − 2x².
3.Differentiate and set to zero: A′ = 200 − 4x = 0, so x = 50.
4.Then y = 200 − 2(50) = 100. Since A′ changes + → − at x = 50, it is a maximum.

Answer: The field is 50 m by 100 m, giving a maximum area of 5000 m². The dimension parallel to the river is twice each perpendicular side.

Tip

Always write the constraint equation explicitly and use it to eliminate a variable before differentiating. Optimization only works once the objective is a function of a single variable.

Checkpoint

To find the absolute maximum of a continuous function on the closed interval [a, b], which points must you test?

Checkpoint

For f(x) = x³ − 3x on [0, 3], the critical point in the interval is x = 1. What is the absolute minimum value of f on [0, 3]?

On the exam

On free response you must justify that your answer is a max or min — cite a sign change in f′ (first-derivative test), the sign of f′′ (second-derivative test), or the candidates comparison. An unjustified extremum loses the justification point.

Answer the 2 checkpoints as you read.