find a number c in (1,2)

yma16

such that when xc, it does not.  You can provide an estimate if it is too hard to find c.  The expression is

 

          .

        .

      .

    x

  x

x

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pagoda

let          .www.ddhw.com        x     x   x x    = S taking Logrithmic Ln on both side, we have         .       x     x   x x  ln(x) = ln(S) or S ln(x) = ln(S)     (1)  www.ddhw.com Solve x = S(1/S)  for the problem is to find the maximum of x in (1,2). In euqation (1) taking derivative w.r.t. S x' = x(1-lnS)/S2, x'=0 yields S = e   so that xmax=e1/e = 1.444668... is the number we are looking for.

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