- Let $f(x,y) = x^2+xy+y^2-3x$.
- Find all $(x,y)$ locations where $\vec \nabla f = \vec 0$.
- At each location you found, does the function have a maximum, minimum, or saddle at that point. Use the eigenvalues of the second derivative to make your decision.
- You can find lots more practice with this in section 14.7 in the text.
- Let $f(x,y) = x^2+y^2$. Your goal is to find the smallest value of $f$ at points on the curve $g(x,y)=xy^2=16$.
- Draw the curve $xy^2=16$. Then add to your plot several level curves of $f$.
- Find $\vec \nabla f$ and $\vec \nabla g$.
- Solve the system $\nabla f=\lambda \nabla g$ together with $xy^2=16$. (Show the three equations you need to solve are $2x=\lambda y^2$, $2y=2\lambda xy$, and $xy^2=16$.)
- Repeat 1. with $f(x,y) = x^3-6x-y^2+4x$.
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