The following graph shows the graph of , as you move the slider the point will move along. The graph of its derivative, can be shown/hidden by clicking the checkbox. What happens to the derivative when g(x) reaches a maximum/minimum?
If a function is continuous on the interval , the function will have an absolute maximum and minimum on that interval, and these points can occur at the endpoints of the interval. A good strategy to find the maximum/minimum of functions is:
- Find the critical points of in (, , ..., ), i.e. where the derivative is zero.
- Evaluate the function at the critical points ,...,,.
- The highest/lowest value found this way will be the maximum/minimum of the function on .
On the next graph, try to identify the absolute maximum of on and . What are the critical points of ? Is defined at 0? What about ?