Prove that function attain its supremum and infimum

Question

I have problem with following task : prove that the function

$ f(x,y,z)=\ln(1+x^2+y^2+z^2)\cos(xyz)e^{-(x^2+y^4+z^6)} $

attains a supremum and infimum.

Answer 1

If $r=\sqrt{x^2+y^2+z^2}$ then we have, for some $C\gt 0$,

$$\tag{*}\label{*}|f(x, y, z)|\le C\ln(1+r^2)e^{-r^2}\to 0 \;\text{when}\; r\to\infty$$

Thus, if you fix any point $(x_0, y_0, z_0)$ where $f(x_0, y_0, z_0)\gt 0$, there will be $R\in\mathbb R$ such that for $r\gt R$ we have $f(x, y, z)\le|f(x, y, z)|<f(x_0, y_0, z_0)$.

Now, the function $f$ reaches its maximum on the disk $r\le R$, because this function is continuous and the disk is compact in $\mathbb R^3$. Thus, it follows that this is also the maximum of $f$ in the whole of $\mathbb R^3$ (due to the previous estimation outside of this disk).

The argument for the infimum/minimum is very similar.

$\eqref{*}$ Note: this is because $e^{-(x^2+y^4+z^6)}=e^{-(x^2+y^2+z^2)}e^{-(y^4-y^2)}e^{-(z^6-z^2)}$, and because $y^2, z^2\ge 0$, the latter two factors are bounded from above, and so is their product - call the upper bound of that product $C$.