<s>[INST] Find all local maxima and minima of $x^2+y^2$ subject to the constraint $x^2+2y=6$. Does $x^2+y^2$ have a global max/min on the same constraint? This is my method for the local max/min. Does this answer sound sensible? (Not sure how to go about checking for global max/min though...)
Method
$G(x,y)=x^2+2y-6$. Rewrite in terms of $y$, so $y=((-x^2+6)/2)^2$. Then $y$ substitute into $f(x)$ so:
$f(x)=x^2+((-x^2+6)/2)^2=-2x^2+(x^4/4)+9$. 
So $f'(x)=-4x+4x^3$. And if $f'(x)=0$ then $x=0,x=1$
And $f''(x)=-4+12x^2$ so when $x=0$ then $f''(x)=-4$ so there is a maximimum when $x=0$.
There is a local maximum at (0,3) with value 9.
When $x=1$ then $f''(x)=8$ so there is a minimum when $x=1$.
There is a local minimum at (1/5/2) with value 29/4.
I also tried answering this question via an alternative method using a Lagrange function, to check my result but I got a different answer...
 [/INST] From $x^2+2y=6 $ we get $ x^2=6-2y$
Substitute in $x^2+y^2$ to get  $$x^2+y^2=y^2-2y+6$$ Note that $y^2-2y+6=(y-1)^2+5$.
Thus the minimum is attained when at $y=1$ at two points where and $x=2$ or $x=-2$
There is no global for $(y-1)^2+5$, hence there is no global maximum for $ x^2+y^2$  subject to $x^2+2y=6 $.
</s>