Sketching curves

I'm having a bit of trouble with this. Any pointers would be greatly appreciated. Thanks

Q) Consider the following statements and determine whether they are true or false. In either case you should provide an example. (i) If f : R to R is a differentiable function that is strictly decreasing everywhere, then f'(x) < 0 for all x in R. (ii) If g : R to R has a maximum at x(subscript 0) = 0 then g'(0) = 0. [R]

Any thoughts of your own on these? Do you think they are true or false?

I presume that the functions can be any function, and there is no requirement for continuity or differentiability, except as mentioned in your post.

Well, for i) I would guess that's false, because from just basic knowledge of curves I know when the first derivative is <0 at a certain x-value it means the function is decreasing at that point, but I'm not sure if you can have an always decreasing function that has f'(x)=0 (inflection point)?

As for ii) I think it could be true but I'm not sure

You've got it. So, can you think of a simple curve with a point of inflection. Polynomials would be good to look at.



Unless there is a requirement to only consider differentiabley functions, then this would actually be false. Difficult to think of a hint, but consider a discontinuous function.

Something like f(x)= -x^3?

And I'm not 100% sure what ii) means

That's the one I was thinking of.



How about something like

$g:\mathbb{R}\to \mathbb{R}$

with g(x)=0, $\forall x\in\mathbb{R}\setminus\{0\}$
and g(0)=1

Here the derivative isn't even defined at x=0, let alone being equal to 0.

Sorry, I'm not very familiar with the notation lol

Sorry, I'm not very familiar with the notation lol

He means a function:

$\displaystyle g(x)= \begin{cases} 0, & x \neq 0 \\ 1, & x = 0 \end{cases}$

It's just defining g(x) to be zero everwhere, execpt when x=0, in which case we define it to be 1. The graph would be a straight line alone the x-axis, except at x=0. As RDKGames so nicely LaTex'ed it - PRSOM.

The maximum is clearly 1, at x=0, but it's not even differentiable there.

Hmm I'm having a bit of difficulty picturing it in my head...

I was going to upload a picture, but either my machine, or TSR, isn't even giving me the option. So, this crappy text will have to do. The x's mark the graph.


--------------------x------------------
---------------------------------------
---------------------------------------
xxxxxxxxxxxxxxx-xxxxxxxxxxxxxx

I was going to upload a picture, but either my machine, or TSR, isn't even giving me the option. So, this crappy text will have to do. The x's mark the graph.


--------------------x------------------
---------------------------------------
---------------------------------------
xxxxxxxxxxxxxxx-xxxxxxxxxxxxxx

Hmm I'm having a bit of difficulty picturing it in my head...

It's literally just this:



and clearly as ghostwalker said, this has a maximum at $x=0$, but it is not differentiable at that point.

Ok, I understand now. Thanks for both of your help

Somehow I think that that part of the question assumes g is differentiable, in which case the statement would be true. But it's not totally clear.