Can someone please explain where I am going wrong in the given argument? Is it obvious that $f(m,s)>0$?

Question

I have a two-variable function $f(m,s)$ for real $m>0$ and $s>0$, and as the first picture shows, it seems that it is always positive; on the other hand, I can easily find the minimum of $f(m,s)$ that I denote by $fmin$ below but the second picture shows that this minimum can be negative. Is not this a contradiction? I will be grateful if someone explains where I am going wrong. If $f(m,s)$ is always positive why its minimum is negative? Is it obvious that $f(m,s)>0$?

q = 1/3 ArcTan[36 m^2 s^2 - s^4, 6 Sqrt[3] m Abs[-4 m^2 s + s^3]]; (*q\[Element] [-Pi/6, Pi/6]*)

f = 3 m^2 + 2 s^2 - s Sqrt[12 m^2 + s^2] (Cos[q] + Sqrt[3] Sin[q]); (* (Cos[q] + Sqrt[3] Sin[q]) \[Element] [0, Sqrt[3]] implying  that  minimum of f  is  as   follows*)

fmin = 3 m^2 + 2 s^2 - s Sqrt[12 m^2 + s^2]  Sqrt[3] ;

{  Plot3D[f, {m, 0, 10}, {s, 0, 10}, PlotRange -> {-5, 5}, AxesLabel -> Automatic, PlotPoints -> 50]   ,   
 Plot3D[fmin, {m, 0, 10}, {s, 0, 10}, PlotRange -> {-5, 5}, AxesLabel -> Automatic, 
  PlotPoints -> 50]  }

Answer 1

q = 1/3 ArcTan[36 m^2 s^2 - s^4, 6 Sqrt[3] m Abs[-4 m^2 s + s^3]]; 

Your statement that q ∈ {-Pi/6, Pi/6} is not correct

SetOptions[#, WorkingPrecision -> 20] & /@ {NMinValue, NMaxValue};

{qmin, qmax} = 
 N[#[{q, m > 0, s > 0}, {m, s}] & /@ {NMinValue, NMaxValue}] // Chop

(* {0, 1.0472} *)

RootApproximant[qmax/Pi]*Pi

(* π/3 *)

Consequently, q ∈ {0, Pi/3}

Plot3D[{q, Pi/3}, {m, 0, 10}, {s, 0, 10},
 PlotRange -> {0, Pi/2},
 PlotStyle -> {Automatic, Opacity[0.2]},
 AxesLabel -> Automatic,
 Mesh -> None,
 PlotPoints -> 50,
 MaxRecursion -> 7]

Assuming[s > 0, Limit[q, m -> 0, Direction -> "FromAbove"]]

(* π/3 *)

Assuming[s > 0, Limit[q, m -> s/2]]

(* 0 *)

Looking at f

f = 3 m^2 + 2 s^2 - s Sqrt[12 m^2 + s^2] (Cos[q] + Sqrt[3] Sin[q]); 

SetOptions[NMinValue, WorkingPrecision -> MachinePrecision];

fmin = NMinValue[{f, m > 0, s > 0}, {m, s}] // Chop

(* 0 *)

Plot3D[f, {s, 0, 10}, {m, 0, 10},
 PlotRange -> {-1, 5},
 AxesLabel -> Automatic,
 PlotPoints -> 50,
 MaxRecursion -> 7,
 ClippingStyle -> None]

Assuming[s > 0, Limit[f, m -> 0, Direction -> "FromAbove"]]

(* 0 *)

Assuming[s > 0, Limit[f, m -> s] // FullSimplify]

(* 0 *)