How about considering a monotonic increasing function such as over the domain? Notice that is maximized when , but . The key here is that the absolute maximum can exist on the boundary in this case, but if the domain is then it won't happen.
The problem reads like this:
Is it possible for a continuous smooth function g(x) on the domain [-6, 6] to have an absolute maximum at a point x_{0} in [-6, 6] without the derivative f'(x_{0}) being 0? If so, draw an example of such a function.
Here are my workings:
I think that it is not posible to have absolute maximum without being f'(x_{0}) = 0. Am I correct?
How about considering a monotonic increasing function such as over the domain? Notice that is maximized when , but . The key here is that the absolute maximum can exist on the boundary in this case, but if the domain is then it won't happen.