# Critical points and constrained domain boundaries

> ML preparation item.

- Stable ID: `mit-18-02sc-f2010-exam2-q3`
- Area: Mathematics
- Topic: Calculus
- Origin: external
- Expected time: 14 minutes
- Skills: critical-points, hessian-test, boundary-optimization
- Item JSON: https://mlprep.iwase.dev/data/items/mit-18-02sc-f2010-exam2-q3.json

This site does not reproduce the third-party problem.

- Source: MIT 18.02SC Multivariable Calculus
- Document: Exam 2
- Locator: Question 3(a–b), page 1
- Official answer PDF page(s): 1
- Original problem: https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/resources/mit18_02sc_exam2/
- Official answer: https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/resources/mit18_02sc_exam2sol/
- Answer status: official-solution


## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `mit-18-02sc-f2010-exam2-q3`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.