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Consider the function $f(x,y) = x^2y+xy^2-2xy$.
- Sketch the level set of $f$ at height $k=0$. Be as accurate as you can, and be sure to label your axes and their important values. Do not use a computer or graphing calculator to do this for you. You should be able to analyze this situation yourself. Make your plot at least $3\times 3$ inches in size. Label your level set as "$k=0$".
(Hint: factor out $xy$.)
- On your plot, shade the region of points where the function's value is positive. Explain how you determined this.
- On the same diagram, sketch what you think the level set at height $k=.1$ should look like. Note that you will not be able to compute it exactly; you should use reasoning not computation to do this. (Do not plot this using a calculator or computer and then copy that; if you do, you are missing the point of the exercise.) Make sure it is clearly labeled in your drawing and that it can be distinguished from your
- On the same diagram, sketch what you think the level set at height $k=-.1$ should look like. Make sure it is clearly labeled in your
- Indicate the (approximate) positions of any critical points for $f$ on your diagram. How did you locate these?