my task is to figure out the critical points of $f(x,y)=e^y(x^4-x^2+y)$, $\ $$\mathbb{R}^2 \rightarrow \mathbb{R}$, and show which of them is a maximum or minimum. As far as I got, I've shown that the critical points are:
1.: $(0,-1)$ which is neither max. nor min. (char. pol. of Hessian is indefinite)
2.: $\left(\frac{1}{ \sqrt2},-\frac{3}{4}\right)$, which is a local minimum and
3.: $\left(-\frac{1}{ \sqrt2},-\frac{3}{4}\right)$ which is the second local minimum.
Moving towards my question, is there any way to easily show if any global maximum or minimum exists (in general and/or concerning this example)? I’ve used the char. poly. of the Hessian, is there any faster possibility to finding local min./max.?
Best Answer
First find the limit as x and y approach infinity. In single variable functions, you only have to check to "ends", $-\infty$ and $\infty$. In functions of two variables, there are four, $x\to-\infty$, $x\to\infty$, $y\to-\infty$, and $y\to\infty$.
$$\lim_{x\to-\infty}f(x,y)=\infty$$
$$\lim_{x\to\infty}f(x,y)=\infty$$
$$\lim_{y\to-\infty}f(x,y)=-\lim_{y\to\infty}\frac{e^{-y}}{\frac{1}{y}}=0$$
$$\lim_{y\to\infty}f(x,y)=\infty$$
Evaluate $f(x,y)$ at the critical points you calculated and compare them to these values.