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Problem Of The Week 43

with 14 comments


Find the number of real roots (x+1)^x=x^(x+1)

Written by Implex

October 4, 2008 at 2:10 am

14 Responses

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  1. No solution!But I couldn’t find any method do prove it..

    Celebrating Life

    October 4, 2008 at 3:34 am

  2. try again !!

    outtimed

    October 4, 2008 at 5:02 am

  3. One solution
    -1

    tyro

    October 4, 2008 at 5:45 am

  4. what is the method?

    outtimed

    October 4, 2008 at 6:12 am

  5. I took log to the base x of both sides and solved it for x.

    tyro

    October 4, 2008 at 6:19 am

  6. But then x can’t be -1. so it could be no solution.

    tyro

    October 4, 2008 at 7:14 am

  7. there is a solution try again !

    outtimed

    October 4, 2008 at 7:23 am

  8. I tried constructing a graph for the equation. Infinite real roots?
    Totally confused with the question. @ Outtimed can you please explain the answer.

    riya

    October 4, 2008 at 10:40 am

  9. The graph is a straight line parallel to an axis

    tyro

    October 4, 2008 at 11:49 am

  10. The value of x or if we think of it as values , will lie between 2 and 3 as for 2 or less than 2 one side of the equation becomes greater than the other and for 3 or greater than 3 , the other side of the equation becomes greater.

    tyro

    October 4, 2008 at 12:46 pm

  11. x comes to be the xth root of (1+1/x). Thus this forms an infinite series.

    tyro

    October 4, 2008 at 2:24 pm

  12. log(x+1)/(x+1)=logx/x

    take f(y)=logy/y
    plot the graph

    now check if you can find a solution or not

    hint: the graph does not rise or dip monotonously!

    outtimed

    October 4, 2008 at 4:24 pm

  13. basically we get

    (1+1/n)^n =n
    so basically we get one root between 2 and 3 and later on as limit of left hand side is e when n approaches infinity.. so just one root..

    Varun

    October 4, 2008 at 6:05 pm

  14. plotting the graph for log y/y between y=2 to 4 we get a curve which first increases then decreases, the value is almost same for 2.3 and 3.3, around 0.157

    tyro

    October 5, 2008 at 3:53 am


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