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plotting - What is my problem with MeshFunction?


I want plot the max value in a sine plot. We can use the following code



Plot[Sin[x], {x, 0, 50}, Mesh -> {{.99}}, 
  MeshFunctions -> {#2 &}, 
  MeshStyle -> {PointSize[Large], Red}]

Why can't Mesh -> 1 be used to plot the red point?


But in a mathematics way, I want to use this code


Plot[Sin[x], {x, 0, 50}, 
  Mesh -> {{1}}, 
  MeshFunctions -> {Boole[(Cos[#] == 0) && (-Sin[#] < 0)] &}, 
  MeshStyle -> {PointSize[Large], Red}]


You can see the MeshFunction doesn't work? Why?


Then I try some other test. For example,I want to emphasize the point above 0.5 to draw on Red. I use this code


Plot[Sin[x], {x, 0, 50}, 
  Mesh -> {{1}}, 
  MeshFunctions -> {Boole[Greater[#2, 0.5]] &}]

It doesn't work again. So I guess the equation PrimePi[z] == 2 will give 2 <= z < 3. To see whether this region can plot in MeshFunction, I tried the following:


f[x_, y_] := (x^2 + 3 y^2)*E^(1 - x^2 - y^2)


Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}, 
  Mesh -> {{1}}, 
  MeshFunctions -> {PrimePi[#3] &}]

Mathematica graphics


You can see the plot is weird.


So what is my problem with MeshFunction? Can I use MeshFunction to plot the maximum value in a plot?



Answer



Another alternative is to use ConditionalExpression using the second-order condition for a local maximum as the second argument:


f = Sin;

Plot[f[x], {x, 0, 20 Pi}, 
     Mesh -> {{0}},
     MeshFunctions -> {ConditionalExpression[f'[#], f''[#] < 0] &},
     MeshStyle -> {PointSize[Large], Red}]

enter image description here


f = Sin[#] - 1/2 Cos[Pi #] &;
Plot[f[x], {x, 0, 10 Pi}, 
     Mesh -> {{0}},
     MeshFunctions -> {ConditionalExpression[f'[#], f''[#] < 0] &},

     MeshStyle -> {PointSize[Large], Red}]

enter image description here


You can add additional constraints to the second argument of ConditionalExpression


e.g. f[#]>0:


 Plot[f[x], {x, 0, 20 Pi},
      Mesh -> {{0}}, 
      MeshFunctions -> {ConditionalExpression[f'[#],f''[#] < 0 && f[#] > 0] &},
      MeshStyle -> {PointSize[Large], Red}]


enter image description here


Update ... or, essentially any constraint:


cond = ((# - 10 Pi)/(2 Pi))^2 + (Pi #2)^2 &;
rplt = RegionPlot[{#, ! #}, {x, 0, 20  Pi}, {y, -2, 2},
                  PlotLegends -> "Expressions"] &@(4 < cond[x, y] < 16);
meshF = ConditionalExpression[f'[#], f''[#] < 0 && ff@(4 < cond[#, f[#]] < 16)] &;
Row[Table[Plot[f[x], {x, 0, 20 Pi},        
                Mesh -> {{0}}, ImageSize -> 400,
                MeshFunctions -> {meshF}, 
                MeshStyle -> {PointSize[Large], Red},                            

                Epilog -> {Opacity[.6], rplt[[1, 1]]}],
         {ff, {Identity, Not}}],
    rplt[[2, 1, 1]]]

enter image description here



See also: this great answer by Silvia for a more general method.


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