I want to find the global extrema of $$f(x,y,z)=x+y+z$$
on the set
$$A=\{(x,y,z) \in \mathbb R^3:x^2+y^2 \le z\le 1\}$$
As $A$ is a compact set and $f$ is a continuous function, it has a minimum and a maximum on $A$. There are no local extrema because the gradient of the function is, at no point, equal to zero. From now on, I have no idea how to approach this problem further. Any help would be appreciated.
since we have $$\frac{\partial f(x,y,z)}{\partial x}=\frac{\partial f(x,y,z)}{\partial y}=\frac{\partial f(x,y,z)}{\partial z}=1$$ you must look for the extrema on the curves $$x^2+y^2=z=1$$ the maximum will be reached for $$(x,y,z)=\left(\frac{1}{\sqrt{2}};\frac{1}{\sqrt{2}};1\right)$$ and the minimum for $$(x,y,z)=\left(-\frac{1}{2};-\frac{1}{2};\frac{1}{2}\right)$$