Interpretation of a Global Minima in $\mathbb{R}^2$

Question

I have been set the following task as an assignment. Given the following function: $$f(x,y) = 4x^3 +y^3 -6xy$$ Find the global minimum in the domain $\mathbb{R}^2$

Am I right in interpreting the domain as $[\infty, -\infty]$ in both variables? The function's value will just continue getting more negative as $x$ and $y$ get more negative.

Presumably, therefore, the function just takes on a value of $-\infty$ at the global minima. Would you agree with this, or have I misunderstood something? It just seems slightly trivial and feels like I have misunderstood something.

Appreciate any advice.

Answer 1

The function has not global maximum nor mimimum since

• for $y=0,x \to +\infty \implies f(x,y)\to +\infty$
• for $y=0,x \to -\infty \implies f(x,y)\to -\infty$

To establish the existence of global maximum and minimum you need to refer to the Extreme value theorem .

Notably the existence of a global maximum and a minimum is guaranteed if the domain of f is compact (i.e. closed and bounded) and the function is continuos. Otherwise the existence is not guaranteed.

Answer 2

There wouldn't be a global minimum, say, $\lim_{x\rightarrow-\infty}f(x,1)=\lim_{x\rightarrow-\infty}(4x^{3}-6x+1)=\lim_{x\rightarrow-\infty}x^{3}\left(4-\dfrac{6}{x^{2}}+\dfrac{1}{x^{3}}\right)$ and $4-\dfrac{6}{x^{2}}+\dfrac{1}{x^{3}}$ is bounded away from zero and strictly positive when $x<<0$, so $\lim_{x\rightarrow-\infty}f(x,1)=-\infty$.