The graph of f' (not f) is given below.

 http://webwork.marianopolis.com:81/wwtmp/Haldane_course/gif/1131348-125-setNYA-Assignment-9prob3image1.png

At which of the marked values of x is

A.f(x) greatest? x=

B.f(x) least? x= ?
C.f'(x) greatest? x= ?
D.f'(x) least? x= ?
E.f''(x) greatest? x=?
F.f''(x) least? x= ?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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.

Get 48 Hours Free Access
Approved by eNotes Editorial