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Thread: Find all the positive roots of a given equation and prove an inequality w/derivatives

  1. #1
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    greece
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    Find all the positive roots of a given equation and prove an inequality w/derivatives

    Given
    $$f(x)=\left\{
    \begin{array}{ll}
    x\cdot \ln(x) & \mbox{if } x > 0 \\
    0 & \mbox{if } x = 0
    \end{array}
    \right.
    $$




    a) Prove that $f(x)$ is continuous at $x=0$.




    b) Study $f(x)$ as for the monotony and find its domain.




    c)
    Find the number of the positive roots of the equation $x=e^{a/x}$ for all the real values of $a$.




    d) Prove: $f'(x+1)>f(x+1)-f(x)$, $\forall x>0$.




    Note:
    Questions a, b have already been solved. I'm struggling to move on to questions c and d.
    a) it is continuous at 0.
    b) f(Df)=$[-(1/e),\infty)$, at $(0,1/e]$ f(x) is a strictly decreasing function whereas at $[1/e,\infty)$ f(x) is a strictly increasing function.
    Regarding c I've thought of using Bolzano's Rule but not exactly sure on how to apply it in the current situation.
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  2. #2
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    Re: Find all the positive roots of a given equation and prove an inequality w/derivat

    (c) this is equivalent to solving

    $$\ln x = \frac{a}{x}$$

    or

    $$f(x)=a$$
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  3. #3
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    Re: Find all the positive roots of a given equation and prove an inequality w/derivat

    Can you please elaborate on your answer?
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  4. #4
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    Re: Find all the positive roots of a given equation and prove an inequality w/derivat

    Quote Originally Posted by Exodia View Post
    Can you please elaborate on your answer?
    which part? taking $ln$ of both sides we get $x ln x=a$ which is $f(x)=a$ for $x>0$

    so we are asked to find the solutions of $f(x)=a$

    we can use the graph of $f$ to answer this question

    Find all the positive roots of a given equation and prove an inequality w/derivatives-graph-f.jpg
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