Proving global minima of function

Question

I have the following function: $$f(x,y,z)=x^2+y^2-\log(x^2y^2\sin^2z)$$ Dealing with the case where $x>0,y>0,z\in(-\pi,\pi]$, I have shown that it has stationary points at $(1,1,-\pi/2)$ and $(1,1,\pi/2)$. Using the Hessian, I've shown that they are local minima. I've shown that the function goes to positive infinity when $x$ or $y$ go to zero or infinity, and when $z$ goes to zero, $-\pi$, or $\pi$. How can I show that these minima are global minima in this region? Is there a general theorem I can use?

Answer 1

$$f(x,y,z)\geq \sum_{cyc}(x^2-2\ln{x})\geq2.$$

Indeed, let $g(x)=x^2-2\ln{x}.$

Thus, $$g'(x)=2x-\frac{2}{x}=\frac{2(x-1)(x+1)}{x},$$ which gives $x_{min}=1$ and $$g(x)\geq g(1)=1.$$

The equality occurs for $(x,y,z)=\left(1,1,\frac{\pi}{2}\right),$ which says that $2$ is a minimal value of $f$.

Answer 2

You can refer to the special case of Weierstrass Extreme value theorem.

Even if the domain is not compact, since the function is continuos and tends to $+\infty$ at the boundary, it must reach a minimum value in the domain.