A big part of my calculus class is using Lagrange multipliers to find max/min values of a given function subject to some constraint. One thing I'm struggling with however, is that the endpoints of your given constraint curve are supposed to be considered as potential maxima/minima. This is trivial for some curves, and quite easy to see doesn't exist for constraints like spheres as they evidently have no endpoints, but there are some curves that I cannot visualize endpoints for.
For example, the constraint $z^{2} = xy +1$. I have a hard time visualizing it, and drawing it out is time consuming. My question is, how can I check for whether this kind of curve indeed has endpoints or not as simply as possible?
The function can be written as
$$z=\pm\sqrt{xy+1}.$$
Obviously, if $xy<-1$ then there is no real $z$. Note that $xy<0$ if the signs are different. This means that our function has limitations only on the second and the fourth quadrant.
For a positive $x$ we have to find those negative $y$'s for which $xy<-1$. So, we have no solution if $$y<-\frac1x.$$
If $x<0$ then $y$ has to be positive and we don't have a solutions if $$y>-\frac1x.$$ (The inequality turned around because we divided by a negative number.)
The following figure depicts in red the domain where there is no solution, and in green where there is solution:
Alpha plots well depicts the situation. For $+\sqrt{xy+1}$ we have
And for $-\sqrt{xy+1}$ my Alpha PRO gives
The moral of my answer was supposed to be that we did not have to use Alpha, since elementary calculations showed the border lines.