Determine whether or not the function is coercive. $$f(x_1,x_2)=e^{x_1^2}+e^{x_2^2}-x_1^{200}-x_2^{200}$$
Intuitively, I know that this function is coercive because, for large enough $x_i$, we have that $e^{x_i^2}$ grows faster than $x_i^{200}$, and so if the norm of $x$ approaches infinity, then the function will also approach infinity. However, this is not rigorous enough to show that the function is coercive.
Could I get some help on proving that this function is coercive?