SAT Optimization Problems: Finding Maximum and Minimum Values Efficiently

Optimization questions often ask: What is the maximum profit? What is the minimum cost? These questions give constraints (restrictions) and ask for the extreme value. The strategy is always the same: express the target quantity as a function of variables, apply constraints, and find where the function reaches its maximum or minimum. Most SAT optimization can be solved algebraically by recognizing the function and finding its extreme value within the constraint range.

For linear functions (like profit = 5x+2y), the maximum/minimum occurs at the corner points of the constraint region. For quadratic functions (like area = x(10-x)), the maximum/minimum occurs at the vertex. Understanding which type of function you have tells you immediately where to look for the extreme value.

Start by identifying the quantity you are optimizing (maximize or minimize this), the constraints (these restrict your variables), and the relationship between variables. Write out the function to optimize and the constraints as inequalities or equations. For example: "Maximize profit = 50x+30y subject to x+2y ≤ 40 and x,y ≥ 0." Once set up correctly, finding the maximum is straightforward.

Practice translating word problems into mathematical setups. Work through three micro-examples: maximizing area given a perimeter constraint, minimizing cost given quantity constraints, and maximizing profit given supply constraints. After 10-15 setups, the translation process becomes automatic.

For linear optimization, test the corner points of the constraint region; the maximum/minimum is at one of them. For quadratic optimization, find the vertex using x=-b/2a, then check whether this vertex is within the constraint range. If the vertex is outside the constraint range, the optimal value is at a boundary point. This two-step process (find critical point, check constraints) handles most optimization problems.

Practice solving 5-6 linear optimization problems and 5-6 quadratic optimization problems. Build familiarity with both types. Most optimization on the SAT is linear or simple quadratic. After practice, solving these becomes routine.

Each optimization problem should take 2-3 minutes: setup (1 minute), optimization (1 minute), answer (30 seconds). Use 10-question daily drills focusing on either linear or quadratic optimization. Build speed separately on each type, then mix them in combined drills.

After reaching fluency (90% accuracy in under 3 minutes per problem), move on. Optimization is a distinct skill and targeted practice builds it faster than mixed practice. Once fluent, optimization problems on full tests will feel straightforward and fast compared to other question types.