Show that $f(x_1,x_2)=e^{x_1^2}+e^{x_2^2}-x_1^2-x_2^2$ is coercive

Question

I am trying to show $$f(x_1,x_2)=e^{x_1^2}+e^{x_2^2}-x_1^2-x_2^2$$ is a coercive function. I considered the inequality $e^a\geq 1+a \ \ \forall a\in\mathbb{R}$, so that
\begin{align}
e^{x_1^2}+e^{x_2^2}-x_1^2-x_2^2&\geq1+x_1^2+1+x_2^2-x_1^2-x_2^2 \\
&=2.
\end{align}

However, from this, I am unable to show that $f$ is coercive by the below definition $$\lim_{\|x\|\rightarrow\infty} f(x_1,x_2)=+\infty.$$

Answer 1

Use $$e^{x^2}\geq1+x^2+\frac{x^4}{2}.$$ By C-S we obtain: $$e^{x_1^2}+e^{x_2^2}-x_1^2-x_2^2\geq2+\frac{x_1^4+x_2^4}{2}\geq2+\frac{(x_1^2+x_2^2)^2}{4}\rightarrow+\infty.$$