In[]:=
thetak1[a1_,a2_,b1_,b2_,thetak2_]:=-((b2-a2)/(b1-a1))*thetak2+(1/(2(b1-a1)))((b2^2-a2^2)+(b1^2-a1^2))​​thetak2[a1_,a2_,b1_,b2_,thetak1_]:=-((b1-a1)/(b2-a2))*thetak1+(1/(2(b2-a2)))((b2^2-a2^2)+(b1^2-a1^2))​​btheta1[a1_,a2_,b1_,b2_]:=(1/(2(b1-a1)))((b2^2-a2^2)+(b1^2-a1^2))​​btheta2[a1_,a2_,b1_,b2_]:=(1/(2(b2-a2)))((b2^2-a2^2)+(b1^2-a1^2))​​bt1[a1_,a2_,b1_]:=(a2^2-(b1-a1)*(b1+a1-2))^(1/2)​​bt2[a1_,a2_,b1_]:=1-((1+(a2(a2-2)-(b1-a1)*(b1+a1)))^(1/2))​​uBB[b1_,b2_,thetab1_,thetab2_]:=-(thetab1-b1)^2-(thetab2-b2)^2​​uBA[a1_,a2_,thetab1_,thetab2_]:=-(thetab1-a1)^2-(thetab2-a2)^2​​PA1[a1_,a2_,b1_,b2_]:=(1/2)*btheta1[a1,a2,b1,b2]*btheta2[a1,a2,b1,b2]​​PA2[a1_,a2_,b1_,b2_]:=(1/2)(btheta1[a1,a2,b1,b2]*btheta2[a1,a2,b1,b2]-thetak1[a1,a2,b1,b2,1]*(btheta2[a1,a2,b1,b2]-1))​​PA3[a1_,a2_,b1_,b2_]:=(1/2)(btheta1[a1,a2,b1,b2]*btheta2[a1,a2,b1,b2]-((btheta1[a1,a2,b1,b2]-1)*thetak2[a1,a2,b1,b2,1]))​​PA4[a1_,a2_,b1_,b2_]:=1-(1/2)((1-thetak1[a1,a2,b1,b2,1])*(1-thetak2[a1,a2,b1,b2,1]))​​EUB1[a1_,a2_,b1_,b2_,thetab1_,thetab2_]:=(1-PA1[a1,a2,b1,b2])*uBB[b1,b2,thetab1,thetab2]+PA1[a1,a2,b1,b2]*uBA[a1,a2,thetab1,thetab2]​​EUB2[a1_,a2_,b1_,b2_,thetab1_,thetab2_]:=(1-PA2[a1,a2,b1,b2])*uBB[b1,b2,thetab1,thetab2]+PA2[a1,a2,b1,b2]*uBA[a1,a2,thetab1,thetab2]​​EUB3[a1_,a2_,b1_,b2_,thetab1_,thetab2_]:=(1-PA3[a1,a2,b1,b2])*uBB[b1,b2,thetab1,thetab2]+PA3[a1,a2,b1,b2]*uBA[a1,a2,thetab1,thetab2]​​EUB4[a1_,a2_,b1_,b2_,thetab1_,thetab2_]:=(1-PA4[a1,a2,b1,b2])*uBB[b1,b2,thetab1,thetab2]+PA4[a1,a2,b1,b2]*uBA[a1,a2,thetab1,thetab2]​​lb2[a1_,a2_,b1_]:=(b1/a1)*a2
In[]:=
N[bt1[1/8,1/4,4/5],4]​​EG1EUB[b2_,thetab1_,thetab2_]:=Piecewise[{{EUB2[1/8,1/4,4/5,b2,thetab1,thetab2],0<b2≤bt1[1/8,1/4,4/5]&&(1/2)≤thetab1≤1&&0≤thetab2≤1},{EUB4[1/8,1/4,4/5,b2,thetab1,thetab2],bt1[1/8,1/4,4/5]<b2≤1&&(1/2)≤thetab1≤1&&0≤thetab2≤1}}]​​EG1EUBb2[b2_,thetab1_,thetab2_]:=D[EG1EUB[b2,thetab1,thetab2],b2]​​EG1EUBb2b2[b2_,thetab1_,thetab2_]:=D[EG1EUB[b2,thetab1,thetab2],{b2,2}]​​Simplify[EG1EUBb2[b2,thetab1,thetab2]]//TraditionalForm​​Simplify[EG1EUBb2b2[b2,thetab1,thetab2]]//TraditionalForm
Out[]=
0.8878
Out[]//TraditionalForm=
6400
3
b2
-4800(2thetab2+1)
2
b2
+(-4320thetab1+8000thetab2+76)b2+2160thetab1+922thetab2-899
2160
b2>0∧40b2≤
1261
∧
1
2
≤thetab1≤1∧0≤thetab2≤1
-
(1600
2
b2
-3200b2-461)(12800000
4
b2
-1280000(16thetab2+15)
3
b2
-19200(540thetab1-1600thetab2-453)
2
b2
+800(12960thetab1-14400thetab2-4933)b2-4451760thetab1+1280000thetab2+1852839)
1382400000
2
(1-4b2)
1261
40
<b2≤1∧
1
2
≤thetab1≤1∧0≤thetab2≤1
Out[]//TraditionalForm=
80
2
b2
9
-
40
9
(2thetab2+1)b2-2thetab1+
100thetab2
27
+
19
540
b2>0∧40b2≤
1261
∧
1
2
≤thetab1≤1∧0≤thetab2≤1
-40960000000
6
b2
+24576000000(2thetab2+5)
5
b2
+7680000(2160thetab1-17600thetab2-14877)
4
b2
-5120000(6480thetab1-21678thetab2-9253)
3
b2
+960000(19440thetab1-41912thetab2-9881)
2
b2
-1920000(2160thetab1-3539thetab2-438)b2+3235509360thetab1-438240000thetab2-1367883079
172800000
3
(4b2-1)
1261
40
<b2≤1∧
1
2
≤thetab1≤1∧0≤thetab2≤1
In[]:=
Solve[EG1EUBb2[b2,thetab1,thetab2]==0&&(1/4)<b2<thetab2,b2,Reals,Assumptions(1/2)≤thetab1≤1&&0≤thetab2≤1]
Out[]=
b2
Root[-899+2160thetab1+922thetab2+(76-4320thetab1+8000thetab2)#1+(-4800-9600thetab2)
2
#1
+6400
3
#1
&,2] if thetab2>
1
2

In[]:=
Solve[EG1EUBb2[b2,thetab1,thetab2]==0&&(1/4)<b2<thetab2,b2,Reals,Assumptions(1/2)≤thetab1≤1&&(1/2)≤thetab2≤1]
Out[]=
{{b2Root[-899+2160thetab1+922thetab2+(76-4320thetab1+8000thetab2)#1+(-4800-9600thetab2)
2
#1
+6400
3
#1
&,2]}}
In[]:=
b2[thetab1_,thetab2_]:=Root[-899+2160thetab1+922thetab2+(76-4320thetab1+8000thetab2)#1+(-4800-9600thetab2)
2
#1
+6400
3
#1
&,2]
SetDelayed
:Tag Root in Root-899+2160thetab1+922thetab2+(76-4320thetab1+8000thetab2)#1+(-4800-9600thetab2)
2
#1
+6400
3
#1
&,2[thetab1_,thetab2_] is Protected.
Out[]=
$Failed
In[]:=
NSolve[EG1EUBb2[b2,0.9,0.9]==0&&(1/4)<b2<0.9,b2,Reals]
NSolve
:NSolve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.
Out[]=
{{b20.676065}}
In[]:=
b2s[thetab1,thetab2]
Root
:{{b2Root[-899+2160thetab1+922thetab2+Plus[3]Slot[1]+Plus[2]Power[2]+6400Power[2]&,2]}} is not a pair of non-empty lists of equal length.
Out[]=
Root[{{b2Root[-899+2160thetab1+922thetab2+(76-4320thetab1+8000thetab2)#1+(-4800-9600thetab2)
2
#1
+6400
3
#1
&,2]}},2]
In[]:=
Solve[EG1EUBb2[b2,thetab1,thetab2]==0,b2,Reals,Assumptions(1/4)<b2<thetab2&&(1/2)≤thetab1≤1&&(1/2)≤thetab2≤1]
Out[]=
{{b2Root[-899+2160thetab1+922thetab2+(76-4320thetab1+8000thetab2)#1+(-4800-9600thetab2)
2
#1
+6400
3
#1
&,2]}}
In[]:=
b2/.{{b2Root[-899+2160thetab1+922thetab2+(76-4320thetab1+8000thetab2)#1+(-4800-9600thetab2)
2
#1
+6400
3
#1
&,2]}}
Out[]=
{Root[-899+2160thetab1+922thetab2+(76-4320thetab1+8000thetab2)#1+(-4800-9600thetab2)
2
#1
+6400
3
#1
&,2]}
In[]:=
b2/.{b2Root[-899+2160thetab1+922thetab2+(76-4320thetab1+8000thetab2)#1+(-4800-9600thetab2)
2
#1
+6400
3
#1
&,2]}
Out[]=
Root[-899+2160thetab1+922thetab2+(76-4320thetab1+8000thetab2)#1+(-4800-9600thetab2)
2
#1
+6400
3
#1
&,2]