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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.

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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.

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