Stationary points in local optimization

[Yankel]

Hello again,

I have a small problem. I am looking for local minimum and maximum points of the function:

\[f(x,y)=3x^{2}y+y^{3}-3x^{2}-3y^{2}+2\]

The first question was how many stationary points are there. I have found the derivatives by x and y:

\[f_{x}=6xy-6x\]

\[f_{y}=3x^{2}+3y^{2}-6y\]

and compared them to 0. I found 3 points: y=0,1,2.

According to the attached answers, there should be 4. There is either a mistake in the answer attached, or I am missing a point. Can you help me solve the system of two equations please to find all the points ? Thank you !

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Ok, I couldn't find a DELETE button, so I will answer my own question.

When I put the 3 (!) values of y back in the equations, for one of them, x got 2 values, bringing the sum of points to 4. My mistake.

 

[MarkFL]

We don't allow users to delete threads (which is what would happen to a thread if the first post in it is deleted) because that could potentially cause valuable content to be destroyed if a user decided to delete their thread after getting help because they are trying to keep their professor from finding the thread. :D

Your partials look correct, and from them we obtain:

\(\displaystyle x(y-1)=0\)

\(\displaystyle x^2+(y-1)^2=1\)

And as you found, when from the first equation we take $x=0$, we find \(\displaystyle y=1\pm1\), and for \(\displaystyle y=1\) we find \(\displaystyle x=\pm1\) for a total of 4 points. :D