Finding extreme values of a single variable function

If a function is continuous, its rate of change is non-constant and there are intervals in which it's crescent and others where it's decrescent, a natural conclusion is to expect the rate of change to be zero somewhere. [math]\displaystyle{ f'(x) = 0 }[/math] is a necessary but insufficient condition for a point to be a maximum or a minimum. It's insufficient because there are points where [math]\displaystyle{ f'(x) = 0 }[/math] and yet that point is neither a maximum nor a minimum. The same concept applies to multivariable functions. A point can have a necessary property to be a critical point and yet fail to meet other necessary criteria to be a maximum or a minimum.