Hi.
I tried to see whether this equation is positive, and I couldn't find out the best way.

equation: (-2 a (-2+b)^2 (1+b) c+c^2 (2 b^3+8 [Alpha]-6 b^2 [Alpha]))/(2 (-4+b^2)^2)

• Question:
  What is the correct way to generate a graph with several conditions?

• Condition:
  a>c, c>0, 0<b<1, alpha>1

• Attempted Code:

  RegionPlot[(c (-a (-2+b)^2 (1+b)+c (b^3+4 \[Alpha]-3 b^2 \[Alpha])))/(-4+b^2)^2 <0,{\[Alpha],1,10},{b,0,1},{c,0,3},{a,4,9},PlotLegends->"Expressions"]
  
  Reduce[-((c (-1+\[Alpha]^2) (a (-2+b)^2 (1+b) \[Alpha]+(-4+3 b^2) c (1+\[Alpha]^2)))/((-4+b^2)^2 \[Alpha]^3))<0&&a>0&&0<b<1&&c<a&&\[Alpha]>1,{\[Alpha]},R]] 
  Plot[f[\[Alpha]],{\[Alpha],1,10},PlotRange->All]
  
  (FunctionInterpolation[#1,{a,4.,10.},{b,-2.,2.},{c,0.1,3},{\[Alpha],1,10}]&)[-((c (-1+\[Alpha]^2) (a (-2+b)^2 (1+b) \[Alpha]+(-4+3 b^2) c (1+\[Alpha]^2)))/((-4+b^2)^2 \[Alpha]^3))]
  

I even tried to fix parameters and just want to see how move when alpha is increasing. but it did not show any result. Please help me.

Attempted Code:

a = 10;
c = 3;
b = 0.9;

f(\[Alpha]) = (-2 a (-2 + b)^2 (1 + b) c + c^2 (2 b^3 + 8 \[Alpha] - 6 b^2 \[Alpha]))/(2 (-4 + b^2)^2);

Plot[f[\[Alpha]], {\[Alpha], 1, 9},
  PlotRange -> All,
  Frame -> True,
  AxesLabel -> {"\[Alpha]", "f(\[Alpha])"},
  LabelStyle -> {FontFamily -> "Arial", FontSize -> 12}]