Turn on thread page Beta
    • Thread Starter
    Offline

    1
    ReputationRep:
    This is one of those probably really easy questions that I can't do for some reason... Probably lack of sleep. Can anyone help me? Most of the questions on the sheet are to do with the Mean Value Theorem, but I don't see how that applies...

    Let f be continuous on (a, b) and be differentiable on each of (a, c), (c, b), where
    a < c < b. Show that if f′(x) tends to L as x tends to c then f′(c) exists and equals L.
    Offline

    8
    ReputationRep:
    For any x in (c, b), the mean value theorem shows that

    inf{f '(y) : y in (c, x)} <= [f(x) - f(c)] / (x - c) <= sup{f '(y) : y in (c, x)}

    Letting x -> c from above, the inf and sup both tend to L, so

    [f(x) - f(c)] / (x - c) -> L as x -> c from above

    --

    Similarly,

    [f(x) - f(c)] / (x - c) -> L as x -> c from below
    • Thread Starter
    Offline

    1
    ReputationRep:
    Thankyou!
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: February 26, 2006

University open days

  • University of Lincoln
    Mini Open Day at the Brayford Campus Undergraduate
    Wed, 19 Dec '18
  • University of East Anglia
    UEA Mini Open Day Undergraduate
    Fri, 4 Jan '19
  • Bournemouth University
    Undergraduate Mini Open Day Undergraduate
    Wed, 9 Jan '19
Poll
Were you ever put in isolation at school?
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

Equations

Best calculators for A level Maths

Tips on which model to get

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.