Question Topics :
Although every global minimum is also a local minimum,
it may not be always possible to identify a global minimum
by finding all local minima.
Consider the example given in the first lecture of the
module on "Unconstrained Optimization" (Slide Title: Global Minimum and Local Minimum). For the function
shown there, a local minimum does exist. However, the function is not bounded below and therefore, its global
minimum does not exist!
However, for "Convex Programming Problems", every local minimum
is a global minimum.
In the example you mentioned, the function is unbounded and
goes to $-infty$ as $x_1 ---> -infty$ if $x_2$ is set to zero.
Thus, it does not have a global minimum.