Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use Hessian eigenvalues to distinguish a minimum from a saddle.
- Justify the conclusion "The origin is a strict global minimum with value zero" using the stated assumptions.
Before you start
Gradients and symmetric matrices.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Classify the origin for f(x,y)=x²+xy+y².
Why this math matters
Use Hessian eigenvalues to distinguish a minimum from a saddle. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The functions are twice continuously differentiable.
- The Hessian is evaluated at a critical point.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Classify a critical point through second-order geometry
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Classify the origin for f(x,y)=x²+xy+y².
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
∇f=(2x+y,x+2y)=0 only at (0,0)
Solve both first-order conditions simultaneously.
Work through the mathematics
H=[[2,1],[1,2]] has eigenvalues one and three
Positive curvature occurs in every direction.
Check the conclusion
The origin is a strict global minimum with value zero
Completing squares, f=(x+y/2)²+3y²/4, confirms the stronger global conclusion.
The result
The origin is a strict global minimum with value zero
Completing squares, f=(x+y/2)²+3y²/4, confirms the stronger global conclusion.
Common mistakes to catch
- A critical point need not be an extremum.
- A local Hessian test alone normally gives local information.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Classify the origin for x²−y².
Show a hint
Inspect opposite signs in the Hessian.
Reveal answer and explanation
A saddle point
Moving along x increases the value while moving along y decreases it.
Practice 2
Does a zero Hessian determinant settle the classification?
Show a hint
Higher-order terms may matter.
Reveal answer and explanation
No
The second-derivative test is inconclusive in such a degenerate case.
Take the idea with you
Check whether a proposed optimization equilibrium resists small perturbations in every direction.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Optimize while staying on a constraint
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.