Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use parallel gradients to find constrained extrema.
- Justify the conclusion "Maximum two at (1,1), minimum minus two at (−1,−1)" using the stated assumptions.
Before you start
Gradients and circles.
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
Maximize x+y subject to x²+y²=2.
Why this math matters
Use parallel gradients to find constrained extrema. 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 constraint is an equality.
- The circle is the entire feasible set.
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.
Optimize while staying on a constraint
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Maximize x+y subject to x²+y²=2.
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=(1,1)=λ(2x,2y)
At a regular constrained extremum, the objective gradient is normal to the constraint tangent.
Work through the mathematics
x=y and 2x²=2, so candidates are (1,1),(−1,−1)
Solve the multiplier relation together with the original constraint.
Check the conclusion
Maximum two at (1,1), minimum minus two at (−1,−1)
The compact circle ensures extrema exist; comparing both candidates identifies them.
The result
Maximum two at (1,1), minimum minus two at (−1,−1)
The compact circle ensures extrema exist; comparing both candidates identifies them.
Common mistakes to catch
- Do not solve only the gradient equations and forget the constraint.
- Multipliers produce candidates, not automatic maximum labels.
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
What bound proves the maximum directly?
Show a hint
Apply Cauchy–Schwarz to (1,1)·(x,y).
Reveal answer and explanation
x+y≤2
The product of the two norms is √2·√2=2.
Practice 2
Why check the constraint gradient is nonzero?
Show a hint
The regular multiplier argument needs a tangent normal.
Reveal answer and explanation
Singular constraint points need separate analysis
The usual necessary condition can fail to cover them.
Take the idea with you
Optimize a response while maintaining a fixed total squared input.
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: Integrate a radial quantity over a disk
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