Lets begin by examiming the graph of `f'(x)` . `f'(x)>0` for every x that is marked. This indicates that `f(x)` is increasing on the entire interval.

Thus `f(x)` is greatest at `x_6` and least at `x_1` .

We can read off the graph that `f'(x)` is greatest at `x_3` and least at `x_5` .

Finally, `f'(x)` has a local minimum at `x_1` , so `f''(x_1)=0` . Also `f'(x)` has a local maximum at `x_3` , and another local minimum at `x_5` indicating that `f''(x_3)=f''(x_5)=0` .

`f'(x)` is increasing at `x_2` and `x_6` and decreasing at `x_4` . Since `f'(x)` is decreasing at `x_4,f''(x_4)<0` and is the least value for `f''(x)` .

Of the two points where `f'(x)` is increasing, it appears to be increasing faster at `x_6` so `f''(x)` is greatest at `x_6`

-------------------------------------------------------------

**f(x) greatest at `x_6` , least at `x_1` **

**`f'(x)` greatest at `x_3` , least at `x_5` **

**`f''(x)` greatest at `x_6` , least at `x_4` **

------------------------------------------------------------

## See eNotes Ad-Free

Start your **48-hour free trial** to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Already a member? Log in here.

**Further Reading**