Once you have a single-variable objective function, the next step is to find the interval of $x$-values that make physical sense in the problem’s context. This domain is almost always a closed interval, so the Extreme Value Theorem applies: absolute extrema will occur either at critical points inside the interval or at the endpoints of the interval.
A common mistake is using the algebraic domain of the objective function instead of the contextual domain. For example, a quadratic objective function has an algebraic domain of all real numbers, but length cannot be negative or larger than the total amount of material available. To find the contextual domain:
Require all original variables to be non-negative (zero is allowed for endpoints)
Write inequalities for each variable, then solve for your single variable to get bounds
Confirm the final domain is a closed interval
Exam tip: 跳过定义域建立步骤会扣分，即使你的导数计算正确
4. Finding Optimal Values and Interpreting Results★★★★☆⏱ 5 min
The final step of the optimization process is testing to find the absolute maximum or minimum, then answering the question asked. For a closed interval domain, follow this process:
Compute the first derivative of the objective function
Find all critical points that lie inside the domain
Evaluate the objective function at every interior critical point and both endpoints
Select the largest value for a maximum, or the smallest for a minimum, then find any other requested values using the constraint
If the domain is open (e.g., $r>0$ for a radius) and there is only one critical point, you can use the second derivative test: if $f''(c) < 0$ for all $x$ in the domain, the critical point is an absolute maximum; if $f''(c) > 0$, it is an absolute minimum. This works for most open-domain AP optimization problems.
Common Pitfalls
Why: Students default to memorized full perimeter formulas instead of reading that one side needs no fencing.
Why: Students forget that length cannot be negative, and context restricts variables to a physically meaningful interval.
Why: Students assume the critical point must be the extremum and forget the Extreme Value Theorem requires checking endpoints.
Why: Students stop early after finding the critical point and do not confirm what the question asks for.
Why: Students confuse local and absolute extrema, and AP requires explicit justification of absolute extrema for optimization.
Why: Students pick the first variable to solve for without checking which is easier.