In the Lagrange Multipliers Method the points obtained will be critical points (solutions of an equation which have the form $\nabla f(x)=\lambda\varphi(x)$) of an objective function $f$ (of class $C^1$) restrict to a region $M$ which have the form $M=\varphi^{-1}(c)$, where $\varphi$ is a function (of classe $C^1$) that comes from the constraint (which have the form $\varphi(x)=c$).

Usually, the existence of the maximum and the minimum comes from the continuity of $f$ and the compactness of $\overline{M}$. In this case, $f$ have at least two critical points on $\overline{M}$. However, there are cases in which the equation $\nabla f(x)=\lambda\varphi(x)$ give us only one solution $p\in M$ (this is because the other critical point is in $\overline{M}\setminus M$). Here is a possible approach that sometimes works for these cases:

You can see examples of this case here, here and here.