\[ \renewcommand{\C}{\mathbb{C}} \renewcommand{\R}{\mathbb{R}} \renewcommand{\Q}{\mathbb{Q}} \renewcommand{\N}{\mathbb{N}} \renewcommand{\Z}{\mathbb{Z}} \renewcommand{\bar}{\overline} \renewcommand{\Lin}{\mathrm{Lin}} \]

# Instructions

Please record and submit a video of up to 4 minutes (240 sec) in which you walk the viewer through the solution of the problem given.

- Imagine that you are talking to someone that has read the book, but does not know how to actually carry out the solution of the problem.
- Clearly state what facts you are using.
- You may avoid carrying out computations live. Probably this is a good idea given the restricted time available.
- If you do not do computation, make sure to prepare them in advance and state clearly what the numbers represent. For example say: the gradient of \(f(x,y)\) is given by the vector whose components are the partial derivatives. We compute the partial derivatives in \(x\) and in \(y\) to obtain the following expressionâ€¦.
- Your video may be shorter than 240 seconds. It may not be longer than 240 seconds.

Possible ideas include:

- Record yourself in front of a blackboard.
- Record yourself talking while screen-recording a slideshow presentation or a PDF.
- Include excerpts (cutouts/screenshots) from the book of facts your are using.
- You can start this video by saying: "In this video we study â€¦ and show that â€¦" i.e. you can state your claim at the beginning.

You are strongly suggested to prepare additional material like a 3D or 2D graph to aid in communicating your ideas.

Please submit under "File Drop" in Collab.

You will get full marks if you solve the problem and provide a complete explanation for the solution. If you are unsatisfied with your graded result you will have an opportunity to resubmit.

These assignments are in preparation for the second midterm that is going to be an oral exam.

# 1

Study the function

\begin{equation*} f(x,y)= x^2 e^{-x^2-y^2} \end{equation*}on

\begin{equation*} \Big\{[x;y]\in\R^2 \colon \;\|[x;y]\|\leq 2\Big\} \end{equation*}for global and local maxima and minima, and classify all critical points.

Be sure to consider what happens on the boundary. You do not need to use Lagrange multipliers but you can.