Find the extrema of $F(x,y) = x^2 + xy + x + y^2$ on the set $S=\lbrace (x,y): x^2+y^2\le9$$\rbrace$.
a): Find all critical points on the interior of $S$
b). Parameterize the boundary of $S$ in terms of theta using sine and cosine,then write $f$ in terms of theta on the boundary.(Be careful to put restrictions on theta).
c). Find the extrema of $F$ on $S$.
I'm not sure how to do any of the parts for this question, but I know you have to do something with the first derivative and second derivative tests for part A, but honestly I have no idea. In a single variable function, I know you can simply do 1st and 2nd derivative tests to find maxima and minima, but how do I do the above problems in multi variable functions.
For a) solve the System given by $$\frac{\partial F(x,y)}{\partial x}=2x+y+1=0$$ and $$\frac{\partial F(x,y)}{\partial y}=x+2y=0$$