Does the following function admit a maximum? I should determine if a maximum value for the function exist and, if it is so, the maximum points.
$$f(x, y, z) = (1 + x^
2
)e^{-z}$$
over the region:
$$D = \{(x, y, z) ∈ R^
3
|x = y^
2 + z^
2
, x^2 − x + y^
2 + z^
2 ≤ 2\}$$
The set D is clearly closed but I am not sure how to prove that it is compact here. Thank you..

Your region $D$ consists of the intersection of the paraboloid $x=y^2+z^2$ and the interior of the sphere $(x-\tfrac12)^2+y^2+z^2=\tfrac94$ (after completing the square). So $D$ is closed and bounded, and thus compact by Heine-Borel. 
As for the maxima/minima, since the roles of $x$ and $z$ are independent (and there is no $y$ in the formula, you need to maximize/minimize on $x$ and $z$ independently. 
For max: you want $z=0$ and maximize $1+x^2$. In the paraboloid we have $x=y^2+z^2$, so the formula for the sphere because $x^2-x+x\leq 2$, so $x^2\leq 2$. Thus $1+x^2$ maximizes as $1+2=3$ when $x=\pm\sqrt2$. So the maximum is $3$, achieved at points, $(\pm\sqrt2,\pm2^{1/4},0)$ (the values for $y$ coming from $\pm\sqrt2=y^2+0^2=y^2$. 
For the minimum you want $x=0$ to minimize $1+x^2$, and $z$ as big as possible; but from $0=y^2+z^2$ you get $y=z=0$, so the minimum is $1$ at $(0,0,0)$. 
