1. ## Differential Calculus

How about considering a monotonic increasing function such as $\displaystyle f(x)=x^3$ over the domain? Notice that $\displaystyle f(x)$ is maximized when $\displaystyle x_0 = 6$, but $\displaystyle f'(6)=2(6)^2 \neq 0$. The key here is that the absolute maximum can exist on the boundary in this case, but if the domain is $\displaystyle (-6,6)$ then it won't happen.