Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Ishaan94

Let \(x,y\) and \(z\) be positive real numbers less than 4. Prove that among the numbers\[\frac{1}x + \frac1{4-y},\;\frac{1}z + \frac1{4-x},\;\frac{1}y + \frac1{4-z}\]there is at least one that is greater than or equal to 1.

  • one year ago
  • one year ago

  • This Question is Closed
  1. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    Best way to show this, would be to suppose WLOG \[\frac{1}x + \frac1{4-y},\;\frac{1}z + \frac1{4-x}\]are both smaller than 1, and show that this implies \[\frac{1}{y}+ \frac{1}{4-z}\geq1\]

    • one year ago
  2. Ishaan94
    Best Response
    You've already chosen the best response.
    Medals 1

    Thanks. I will try working on it now.

    • one year ago
  3. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    Suppose those first two are smaller than 1. That means \(x>1\) and \(y<(4-x)\). Also, \(z<1\) and \(x<(4-z)\).

    • one year ago
  4. Ishaan94
    Best Response
    You've already chosen the best response.
    Medals 1

    @KingGeorge didn't get the solution :(

    • one year ago
  5. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    I think I messed up on the direction of some of the inequalities. Let me retype that thing real quick.

    • one year ago
  6. Ishaan94
    Best Response
    You've already chosen the best response.
    Medals 1

    Okay.

    • one year ago
  7. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    We need \(x>1\) and \(3>y\) and \(4−y>\frac{x}{x−1}\). We also need \(z>1\), \(3>x\), and \(4−x>\frac{z}{z−1}\). Combine all of these things together, we get the following criteria.\[1<x<3,\qquad x<4−\frac{z}{z−1}\]\[y<4−\frac{x}{x−1},\qquad y<3\]\[1<z\]And various rearrangements. We want to show one of the following. \[y≤1\]\[ 4−z\leq \frac{y}{y−1}\] There we go. I hope that's finally right.

    • one year ago
  8. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    and \(z<4\) of course.

    • one year ago
  9. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    Another way to finish the proof would be to show \(z\geq3\).

    • one year ago
  10. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    Since you can rearrange the topmost row to get \(z>5/2\), you can also now finish by showing that \[\frac{5}{2}\leq \frac{y}{y-1}\]

    • one year ago
  11. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    In other words, if \(y\leq 5/3\), we are done.

    • one year ago
  12. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    Hence, we must choose a \(y>5/3\). Using this, we find that \(x>7/4\), and \(z>9/5\). Therefore, \(y>11/6\), \(x>13/7\), and \(z>15/8\).

    • one year ago
  13. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    In general, if you continue to do this process off to infinity (fairly straightforward to show), eventually you will find that \(x\geq2\), \(y\geq2\), and \(z\geq2\).

    • one year ago
  14. Ishaan94
    Best Response
    You've already chosen the best response.
    Medals 1

    I found another way, sum up the three terms and assume their sum to be less than 3. Proof by contradiction.

    • one year ago
  15. KingGeorge
    Best Response
    You've already chosen the best response.
    Medals 1

    That is also an excellent way one could do this.

    • one year ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.