How to determine coercive functions

Question

A continuous function $f(x)$ that is defined on $R^n$ is called coercive if $\lim\limits_{\Vert x \Vert \rightarrow \infty} f(x)=+ \infty$.

I am finding it difficult to understand how the norm of these functions are computed in order to show that they are coercive.
$a) f(x,y)=x^2+y^2 \\b)f(x,y)=x^4+y^4-3xy\\c)f(x,y,z)=e^{x^2}+e^{y^2}+e^{z^2}$

To show that they are coercive I have to show that as norm goes to infinity the function too should go to infinity right?

Answer 1

Consider the first function $f(x,y) = x^2 + y^2$. This function can be written in terms of vectors as $f(\mathbf{x}) = \|\mathbf{x}\|^2$. Now you can see that $f(\mathbf{x}) \to \infty$ as $\|\mathbf{x}\| \to \infty$.

Here is a hint for the second function. Use the inequality $-\frac{3}{2}(x^2 + y^2) \leq -3xy$ to derive a lower bound for $f(x,y)$. Show that for any $M > 0$ there exists a number $K > 0$ such that $f(x,y) > M$ whenever $\sqrt{x^2 + y^2} > K$.

Answer 2

For (c), use $e^x \ge 1+x$, so $e^{x^2} \ge 1+x^2$.