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Math Help - root of equation

  1. #1
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    root of equation

    hello how would i show that the equation lnx = 4-x has one real root..

    am not sure what to do with the lnx function
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  2. #2
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    You want the zeros of f(x) = \ln x + x -4, 0<x<+\infty. Observe by differentiation that f is strictly increasing. Therefore if f has at most one root, since if a<b were two distinct roots we could chuse a<c<b, and 0=f(a)<f(c)<f(b)=0, a contradiction. To see that one root exists, evaluate f near zero to get a negative value and then above 4 to get a positive value and use the IVT.
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  3. #3
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    Hello, sigma1!

    How about a graphing approach?


    How would i show that the equation \ln x \:=\: 4-x has one real root?

    Graph the functions: . \begin{Bmatrix}y &=& \ln x \\ y &=& 4-x \end{Bmatrix} . and note their intersection(s), if any.
    Code:
            |
          * |
           4o
            | *                     *
            |   *            *
            |     *     *
            |       ◊
            |    *    *
        - - + -o- - - - o - - - - - - - - - -
            | *1        4  *
            |                 *
            |*
            |
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