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

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= ?

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### 1 Answer

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`

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**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` **

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