# Multivariable Calculus question, critical points and boundness.

I'm doing an exercise from the book Multivariable Real Analysis by Kolk and Duistermaat, however I'm not sure how to proceed after some point. It asks the following: we have $$f:\mathbb{R}^{2}\rightarrow\mathbb{R}$$ defined by $$f(x)=x_1^2+x_2^2(1+x_1)^3$$. Show that $$0$$ is the only critical point of $$f$$ and that it is a minimum (Already done). And the second part says to prove that $$f$$ is unbounded from below and above. From viewing the equation and the hypotheses I think the function should be bounded below contrary to what they're asking me to prove, plotting the graph in wolfram alpha shows a surface that look bounded below. Anyway, I wasn't able to prove that it is unbounded above, I tried manipulating the matrix of second derivatives but I didn't get far.

I appreciate any insights.

Freeze the $$x_2$$ value to anything nonzero. Then $$x_2^2>0$$ is fixed so $$f$$ can be viewed as a function of only $$x_1$$. When you take the limit as $$x_1 \to -\infty$$, the $$x_1^2$$ term goes to $$+\infty$$, while the cubic term goes to $$-\infty$$. However, the cubic will dominate the square in the limit.
For showing $$(0,0)$$ is the only critical point, set both partials equal to $$0$$ and solve the simultaneous equations.