Can't find the correct root for this equation

Question

So I have this expression

Expr= (5.70129*10^14 f^2 (7.97764*10^8 + 4 f^2 \[Pi]^2) + 
   6198.06 (-6.34731*10^8 + 4 f^2 \[Pi]^2)^2 (7.97764*10^8 + 
      4 f^2 \[Pi]^2) + 
   13149.6 (6.34731*10^8 + 4 f^2 \[Pi]^2) (3.14945*10^10 f^2 + 
      0.648892 (-7.97764*10^8 + 
         4 f^2 \[Pi]^2)^2))/((3.14945*10^10 f^2 + 
     0.648892 (-7.97764*10^8 + 4 f^2 \[Pi]^2)^2) (2.50582*10^10 f^2 + 
     0.272416 (-6.34731*10^8 + 4 f^2 \[Pi]^2)^2))

I want to solve

 Eqn = Abs[Expre - 0.00012] = 0.99*0.00012

However when I do

Eqn = Abs[Expre - 0.00012] = 0.99*0.00012
aux = Quiet[FindRoot[Eqn, {f, 3000}]];
result = f /. aux[[1]]

I obtain result = 42013.8 instead of a value around 3000 or 4000 (the correct answer).

What am I doing wrong?

Answer 1

We plot the function and find that there is not root for 0.99 in the interval {3000,4000}.

expr[f_] := (5.70129*10^14 f^2 (7.97764*10^8 + 4 f^2 \[Pi]^2) + 
     6198.06 (-6.34731*10^8 + 4 f^2 \[Pi]^2)^2 (7.97764*10^8 + 
        4 f^2 \[Pi]^2) + 
     13149.6 (6.34731*10^8 + 4 f^2 \[Pi]^2) (3.14945*10^10 f^2 + 
        0.648892 (-7.97764*10^8 + 
            4 f^2 \[Pi]^2)^2))/((3.14945*10^10 f^2 + 
       0.648892 (-7.97764*10^8 + 
           4 f^2 \[Pi]^2)^2) (2.50582*10^10 f^2 + 
       0.272416 (-6.34731*10^8 + 4 f^2 \[Pi]^2)^2));
Plot[Abs[expr[f] - expr[0]] - 0.99 expr[0], {f, 0, 80000}, 
 AxesOrigin -> {0, 0}]

FindRoot[0.99 expr[0] - Abs[expr[f] - expr[0]], {f, 6000}]

{f -> 42013.8}

the function seems to be a monotonic function,it maybe easy to handle. We can also change 0.99 to 0.6 and test the FindRoot.