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I have the following inequality

$F(\theta)=2e^{-\theta}[\cos\theta + \phi\sin\theta] - e^{-2\theta}[\cos2\theta + \phi \sin2\theta]>1 $.

Herre $\theta\ge0$ is real variable and $\phi$ is a real constant, say 100. How should I proceed to find all the points such that $F(\theta)$ is greater than 1? How to find the maximum values of $F(\theta)$ as a function of $\theta$?

F[x_]=2Exp[-x](Cos[100*x] + Sin[100*x]/100) - Exp[-2x](Cos[2*100*x] + Sin[2*100*x]/100)
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  • 2
    $\begingroup$ Reduce[F[x] > 1, x]? Or Reduce[F[x] > 1, x, Reals]? $\endgroup$
    – Michael E2
    Commented Jan 10, 2018 at 5:10
  • $\begingroup$ Plotting the function shows that it is highly oscillatory with large amplitude for negative x. $\endgroup$
    – bbgodfrey
    Commented Jan 10, 2018 at 5:12
  • $\begingroup$ The solution is unbounded as x approaches -Infinity. $\endgroup$
    – bbgodfrey
    Commented Jan 10, 2018 at 5:20
  • $\begingroup$ Sorry, forgot to mention that x is greater than or equal to zero. $\endgroup$
    – H. Kenan
    Commented Jan 10, 2018 at 5:32

2 Answers 2

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This is basically a duplicate of 5663. Here I apply a variant of the NDSolve answer to your question:

{max, one} = {"Maximum", "One"} /. Last @ Reap[
    NDSolveValue[
        {
        v'[x]==F'[x],
        v[0]==F[0],
        WhenEvent[v'[x]<0,Sow[x, "Maximum"]],
        WhenEvent[v[x]==1, Sow[x,"One"]]
        },
        v,
        {x, 0, 1}
    ],
    _,
    Rule
];

Here is a plot showing the maximum, and the points where F[x] is 1:

Plot[
    F[x],
    {x, 0, 1},
    Epilog -> {
        Red, Point[Thread[{max, F[max]}]],
        Blue, Point[Thread[{one, F[one]}]]
    }
]

enter image description here

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  • $\begingroup$ Thank you very much. Can I find the numerical values of those red dots (maxima )? If not all, at least some of them? $\endgroup$
    – H. Kenan
    Commented Jan 10, 2018 at 6:03
  • $\begingroup$ @user149973 If you run my code, than the numerical values of the x coordinates are contained in the max variable. $\endgroup$
    – Carl Woll
    Commented Jan 10, 2018 at 6:05
  • $\begingroup$ Thanks a lot! I tried to hit the upvote. But my score is not enough to do it. $\endgroup$
    – H. Kenan
    Commented Jan 10, 2018 at 6:07
  • $\begingroup$ thanks for the help. Little more help. If I change cos(x) and sin(x) to hyperbolic cosh(x) and sinh(x) and run the code, I find an error: "....should be a pair of numbers, or a Scaled or Offset form". Could you please help to sort it out? $\endgroup$
    – H. Kenan
    Commented Jan 11, 2018 at 20:30
  • $\begingroup$ @user149973 - look at the derivative of the modified function F2[x], Assuming[x > 0, Reduce[F2'[x] < 0, x, Reals] // Simplify] evaluates to True so the function is monotonically decreasing for x > 0Then the maximum occurs at x == 0 F2[0] is 1 $\endgroup$
    – Bob Hanlon
    Commented Apr 9, 2018 at 3:08
2
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A cheat using MeshShading for F(x)>1:

p = Plot[F[x], {x, 0, 1}, GridLines -> {None, {1}}, 
   MeshFunctions -> (#2 &), Mesh -> {{1}}, 
   MeshShading -> {Red, Black}];
pts = p[[1, 1, 1]];
lns = Cases[p[[1]], {Black, Line[x__]} :> x, Infinity];
gl = MinMax[#[[All, 1]]] & /@ (pts[[#]] & /@ lns)
Plot[F[x], {x, 0, 1}, MeshFunctions -> (#2 &), Mesh -> {{1}}, 
 MeshShading -> {Red, Black}, GridLines -> {Flatten[gl], {1}}, 
 GridLinesStyle -> Blue]

Approximate end-points:

{{2.04082*10^-8, 0.0156412}, {0.0477311, 0.0621775}, {0.0635002, 
  0.0778018}, {0.111333, 0.124225}, {0.127114, 0.139847}, {0.17508, 
  0.186129}, {0.190876, 0.201735}, {0.239076, 0.247788}, {0.254898, 
  0.263366}, {0.303622, 0.308897}, {0.319546, 0.324378}}

enter image description here

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