Results 1 to 3 of 3

Thread: find number of positive roots of x = e^(ax for all real values of "a"

  1. #1
    New Member
    Join Date
    Mar 2018
    Posts
    2

    find number of positive roots of x = e^(ax for all real values of "a"

    Questions and and b have already been answered. regarding c and d I'm kind of confused on how to get started with them. any suggestions?
    Καταγραφή.jpg
    Last edited by stapel; 03-08-2018 at 01:03 PM. Reason: Reinserting original post text.

  2. #2
    New Member
    Join Date
    Mar 2018
    Posts
    2

    find number of positive roots of x = e^(ax for all real values of "a"

    Note: Parts (a) and (b) have already been solved. I'm struggling to move on to parts (c) and (d).



    Given the following function:

    . . . . .[tex]f(x)\, =\, \begin{cases}x\, \cdot\, ln(x) & \mbox{ if }\, x\, >\, 0 \\ 0 & \mbox{ if }\, x\, =\, 0\end{cases}[/tex]

    (a) Prove that [tex]f(x)[/tex] is continuous at [tex]x\, =\, 0.[/tex]

    (b) Study [tex]f(x)[/tex] as for the monotony, and find its domain.

    (c) Find the number of the positive roots on the equation [tex]x\, =\, e^{a/x}[/tex] for all the real values of [tex]a.[/tex]

    (d) Prove [tex]f'(x\, +\, 1)\, >\, f(x\, +\, 1)\, -\, f(x),\, \forall x\, >\, 0.[/tex]




    My answers for parts (a) and (b):

    (a) It is continuous at 0.

    (b) [tex]f(Df)\, =\, \left[-\dfrac{1}{e},\, \infty\right),[/tex] at [tex]\left(0,\, \dfrac{1}{e}\right]\, f(x)[/tex] is a strictly decreasing function whereas at [tex]\left[\dfrac{1}{e},\, \infty\right)\, f(x)[/tex] is a strictly increasing function.
    Attached Images Attached Images
    Last edited by stapel; 03-08-2018 at 01:04 PM. Reason: Typing out the text in the graphic.

  3. #3
    Elite Member
    Join Date
    Jun 2007
    Posts
    17,869
    Quote Originally Posted by Exodia View Post
    Note: Parts (a) and (b) have already been solved. I'm struggling to move on to parts (c) and (d).



    Given the following function:

    . . . . .[tex]f(x)\, =\, \begin{cases}x\, \cdot\, ln(x) & \mbox{ if }\, x\, >\, 0 \\ 0 & \mbox{ if }\, x\, =\, 0\end{cases}[/tex]

    (a) Prove that [tex]f(x)[/tex] is continuous at [tex]x\, =\, 0.[/tex]

    (b) Study [tex]f(x)[/tex] as for the monotony, and find its domain.

    (c) Find the number of the positive roots on the equation [tex]x\, =\, e^{a/x}[/tex] for all the real values of [tex]a.[/tex]

    (d) Prove [tex]f'(x\, +\, 1)\, >\, f(x\, +\, 1)\, -\, f(x),\, \forall x\, >\, 0.[/tex]




    My answers for parts (a) and (b):

    (a) It is continuous at 0.

    (b) [tex]f(Df)\, =\, \left[-\dfrac{1}{e},\, \infty\right),[/tex] at [tex]\left(0,\, \dfrac{1}{e}\right]\, f(x)[/tex] is a strictly decreasing function whereas at [tex]\left[\dfrac{1}{e},\, \infty\right)\, f(x)[/tex] is a strictly increasing function.
    You are asked to find roots of:

    y = x - e^(a/x)

    Assume some value of 'a' (= 0, 1, 2 etc) and plot the function. Look at the nature of the function.

    Tell us what you found....
    Last edited by stapel; 03-08-2018 at 01:05 PM. Reason: Copying typed-out graphical content into reply.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

Tags for this Thread

Bookmarks

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •