Turn on thread page Beta
    • Thread Starter
    Offline

    13
    ReputationRep:
    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]
    • Study Helper
    Online

    15
    Study Helper
    (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.
    • Thread Starter
    Offline

    13
    ReputationRep:
    (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
    • Study Helper
    Online

    15
    Study Helper
    (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.
    • Thread Starter
    Offline

    13
    ReputationRep:
    (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
    • Study Helper
    Online

    15
    Study Helper
    (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.
    • Thread Starter
    Offline

    13
    ReputationRep:
    (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
    • Community Assistant
    Offline

    20
    ReputationRep:
    Community Assistant
    (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}
    • Study Helper
    Online

    15
    Study Helper
    (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.
    • Thread Starter
    Offline

    13
    ReputationRep:
    (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
    • Thread Starter
    Offline

    13
    ReputationRep:
    (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...
    • Study Helper
    Online

    15
    Study Helper
    (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
    • Thread Starter
    Offline

    13
    ReputationRep:
    (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
    • Community Assistant
    Offline

    20
    ReputationRep:
    Community Assistant
    (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.
    • Thread Starter
    Offline

    13
    ReputationRep:
    (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
    Online

    17
    ReputationRep:
    (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.
    • Study Helper
    Online

    15
    Study Helper
    (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.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: November 3, 2017
Poll
Is the Big Bang theory correct?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.