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    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= real number]
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    (Original post by jordanwu)
    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= real number]
    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.
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    (Original post by ghostwalker)
    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
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    (Original post by jordanwu)
    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)?
    You've got it. So, can you think of a simple curve with a point of inflection. Polynomials would be good to look at.

    As for ii) I think it could be true but I'm not sure
    Unless there is a requirement to only consider differentiable functions, then this would actually be false. Difficult to think of a hint, but consider a discontinuous function.
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    (Original post by ghostwalker)
    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
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    (Original post by jordanwu)
    Something like f(x)= -x^3?
    That's the one I was thinking of.

    And I'm not 100% sure what ii) means
    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.
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    (Original post by ghostwalker)
    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
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    (Original post by jordanwu)
    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}
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    (Original post by jordanwu)
    Sorry, I'm not very familiar with the notation lol
    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.
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    (Original post by RDKGames)
    He means a function:

    \displaystyle g(x)= \begin{cases} 0, & x \neq 0 \\ 1, & x = 0 \end{cases}
    Ah ok I see, thanks
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    (Original post by ghostwalker)
    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...
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    (Original post by jordanwu)
    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
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    (Original post by ghostwalker)
    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
    Sorry what are the 3 dotted lines? Yeah I think I really need some sort of image lol
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    (Original post by ghostwalker)
    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
    :lol:

    (Original post by jordanwu)
    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.
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    (Original post by RDKGames)
    :lol:



    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
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    (Original post by jordanwu)
    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.
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    (Original post by IrrationalRoot)
    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.
    Agreed, +rep. Not being able to guess the lecturer's intent with the question, I opted to go for what is there, rather than what I think should be there.
 
 
 
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