My textbook ask to find the extrema of $f(x,y) = 2x^2+y^2$ on $x^4-x^2+y^2-5=0$. It uses the lagrangian multipliers to find critic points.. Then it computes the function on these points then says "this is the maximum, this is the minimum".
I can't understand: they could be saddle points! I believe that it is using the weierstrass theorem for compact sets.. In this case the vinculum is a compact set, so it must have minimum and maximum! Is my conclusion correct? In general, how can you be sure that the vinculum is compact?
Edit: my question was general: suppose it is not easy to do that substition.. If you want to use multipliers you have to know that maximum/min points exist.. So compact vinculum could be good! Bit how do you recognize it