proving that a global maximizer does not exist Show that no matter what the value of $a$ is chosen, the function $f(x_1,x_2)=x_1^3-3ax_1x_2+x_2^3$ has no global maximizers. Determine the nature of the critical points of this function for all values of $a$.  
Here $\nabla f=(3x_1^2-3ax_2,-3ax_1+3x_2^2)$
Hessian$(f)=\begin{pmatrix}6x_1&-3a\\-3a&6x_2\\ \end{pmatrix}$  
The critical points I found are $(0,0)$ and $(a,a)$.  


*

*at $(0,0)$: Hessian is indefinite so, $(0,0)$ is a point of inflection.  

*at $(a,a)$: if $a>0$, then Hessian is positive definite, hence a strict local minimizer exists, and if$ a<0 $then Hessian is negative definite hence a strict local maximizer exists.  
Is this correct?
But I do not understand how to prove that a global maximizer does not exist.

Set e.g. $x_2=0$ to get the restriction of the function on the first axis to be 
$$
f(x_1,0)=x_1^3.
$$
It is not bounded from above ($x_1^3\to +\infty$ as $x_1\to +\infty$), then the function $f(x_1,x_2)$ is unbounded from above too, and the global maximum does not exist.
