Turn on thread page Beta
    • Thread Starter
    Offline

    0
    ReputationRep:
     \\ \mbox{Prove that for all real x,} \\

\\ \left|\frac{x+1}{x^2+2x+2}\right  | \leq \frac{1}{2}
    Offline

    0
    ReputationRep:
    (Original post by drmath)
     \\ \mbox{Prove that for all real x,} \\

\\ \left|\frac{x+1}{x^2+2x+2}\right  | \leq \frac{1}{2}

    If you cross-multiply by the denominators, you end up with an inequality which is very easy to show that it is true!
    Offline

    1
    ReputationRep:
    well you can rewrite the bit in the mod signs as


    (x+1+0.5x^2)/(x^2+2x+2) -0.5x^2/(x^2+2x+2)

    can you go from here?

    bah the post above me has an easier method. i was thinking diffrerentiation.
    Offline

    7
    ReputationRep:
    I'm not convinced it's as simple as that, but the substitution u=x+1 and then doing the two cases dependent on the sign of u is ok.
    Offline

    6
    ReputationRep:
     \\ 0 \leq (|y|-1)^2  \\

\\ 0 \leq |y|^2 - 2|y| + 1 \\

\\ 2|y| \leq |y|^2 + 1 \\

\\ \frac{|y|}{|y|^2+1} \leq \frac{1}{2} \\

\\ \left| \frac{y}{y^2+1} \right| \leq \frac{1}{2}

    Now let y = x+1.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: June 8, 2008

University open days

  • Southampton Solent University
    All faculties Undergraduate
    Sun, 18 Nov '18
  • University of Bradford
    All faculties Undergraduate
    Wed, 21 Nov '18
  • Buckinghamshire New University
    All Faculties Postgraduate
    Wed, 21 Nov '18
Poll
Black Friday: Yay or Nay?
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.