How to find the minimum of this function?

Question

-((0.6666666666666666` (r Sqrt[(
        63.49604207872798` - 31.74802103936399` r + 
         1.1` (-2 + r)^(2/3) r)/(
        63.496042078727974` - 31.748021039363987` r + 
         1.` (-2 + r)^(2/3) r)] (-9.` + 2 r - (
          0.02309855258140601` r^3)/(-2 + r)^(1/3)) - 
       2 (1 - 0.034647828872109016` (-2 + r)^(2/3) - 2/r) (-2.25` r + 
          r^3 Sqrt[(
           63.49604207872798` - 31.74802103936399` r + 
            1.1` (-2 + r)^(2/3) r)/(
           63.496042078727974` - 31.748021039363987` r + 
            1.` (-2 + r)^(2/3)
              r)] + (r^3 (2.2737367544323206`*^-13 (-2 + r)^(1/3) + 
               r (-6.349604207872807` - 
                  1.1368683772161603`*^-13 (-2 + r)^(1/3) + 
                  r (5.291336839894001` + (-1.0582673679787984` - 
                    5.551115123125783`*^-17 (-2 + r)^(
                    2/3)) r))))/((-2 + r)^(
             1/3) (63.496042078727974` + (-31.748021039363987` + 
                  1.` (-2 + r)^(2/3)) r)^2 Sqrt[(
             63.49604207872798` - 31.74802103936399` r + 
              1.1` (-2 + r)^(2/3) r)/(
             63.496042078727974` - 31.748021039363987` r + 
              1.` (-2 + r)^(2/3) r)])) + 
       2.25` r^2 (-(0.02309855258140601`/(-2 + r)^(1/3)) + 2/

          r^2 + (-4.547473508864641`*^-13 (-2 + r)^(1/3) + 
             r (12.699208415745616` + 
                2.2737367544323206`*^-13 (-2 + r)^(1/3) + 
                r (-10.582673679788002` + (2.116534735957597` + 
                    1.1102230246251565`*^-16 (-2 + r)^(
                    2/3)) r)))/((-2 + r)^(1/3)
              r (63.496042078727974` + (-31.748021039363987` + 
                  1.` (-2 + r)^(2/3)) r)^2 Sqrt[(
             63.49604207872798` - 31.74802103936399` r + 
              1.1` (-2 + r)^(2/3) r)/(
             63.496042078727974` - 31.748021039363987` r + 
              1.` (-2 + r)^(2/3)
                r)]))))/(-2.` + ((0.13859131548843606` + 
        2.` (-2 + r)^(1/3) - 0.09239421032562405` r) r)/(-2 + r)^(
     1/3)))

Is there a way to find the value of $r$ where the function above is at its minimum?

I am trying to find the minimum value of the function?

How to use Mathematica to find the position of the red circle on the graph?

Thank you.

Answer 1

fis your function. Then:

tab = Table[{r, f}, {r, 2.1, 3, 0.0001}];
min = MinimalBy[tab, Last]

{{2.27, 2.45707}}

ListLinePlot[tab, DataRange -> {2.1, 3}, Epilog -> {Red, PointSize@Medium, Point@min}, 
 PlotTheme -> "Detailed"]

Check with FindMinimum:

FindMinimum[f, {r, 2.1}]

{2.45707, {r -> 2.26997}}

Answer 2

I find that using FunctionDomain can be helpful in cases like this.

limits = FunctionDomain[f[r], r]
(* 2. < r < 2.0001 || 2.0001 < r < 2.00025 || 
 2.00033 < r < 100.46 || 100.46 < r < 154.045 || r > 177.877 *)

Now we can use the third element of limits as a constraint in NMinimize.

NMinimize[{f[r], limits[[3]]}, r]
(* {2.45707, {r -> 2.26997}} *)

Below the results are plotted

Show[
 Plot[f[r], {r, 2, 5}, PlotRange -> {{0, 6}, {0, 20}}, 
  PlotStyle -> Black],
 ListPlot[{{2.26997, f[2.26997]}}, 
  PlotStyle -> {PointSize[Large], Red}]
 ]