Question about Lagrange multipliers and constraints that are disks and circles There are two cases that I need to clarify:
If I have a question that gives me a function $f(x,y)$ and asks me to find the absolute maximum and minimum on $x^2+y^2=1$ will my maximum and minimum value always be on the boundary of the circle (or even if this wasn't a circle but any boundary, will my absolute max and min always occur on the boundary)?
In a second case where I have the same function $f(x,y)$ but this time the constraint is a disk $x^2+y^2\le1$, the absolute maximum and minimum could be on the boundary or within the boundary of the disk, is this conceptually correct or wrong?
Thank you!
 A: If you consider $f(x,y)$ over all of $\mathbb{R}^2$, then it might be the case that it attains its maximum and minimum values on some point  not on the boundary of the circle specified by $x^2 + y^2 = 1$. 
It seems that the question is asking you to specifically find the maximum and minimum values of $f(x,y)$ "over" the circle. In other words, you have a circle on the xy-plane, and your function values $f(x,y)$ represent a surface over the xy-plane, but you ignore all the points on the surface that aren't directly over the boundary of your circle. 
For the second part, you ignore all the points on the surface that aren't directly over the interior + boundary of your circle. 
After ignoring all the points that aren't over the boundary (in the first case) or the interior + boundary (in the second case), then you find the z coordinate of the highest point and lowest points of the surface (which is the same as the maximum and minimum values of $f(x,y)$ subject to the given constraints).
Hopefully you can picture what you are doing when you are finding the maximums and minimums of a function subject to some constraints now.
