(*   10-9-23   nexxt real solution   *)


f0 = b + c x^2 + d x^4
f1 = ( e + f x^2 ) * (a^2-x^2)

cons1 = ( ( f0 /. x -> 1 ) == ( f1 /. x -> 1 ) )
cons2 = ( ( D[f0,x] /. x -> 1 ) == ( D[f1,x] /. x -> 1 ) )
cons3 = ( ( D[f1,x] /. x -> a ) == 0 )

cons4 = ( Integrate[ f0,{x,0,1} ] == 1/4 )
cons5 = ( Integrate[ f1,{x,1,a} ] == 1/4 )


bcdef = ( Solve[ { cons1,cons2,cons3,cons4,cons5 }, {b,c,d,e,f} ] )

f0$s = (  f0 /. bcdef )
f1$s = (  f1 /. bcdef )

     (*  check area is 1  *)

Factor[ Integrate[ f0$s, {x, -1, 1}] + 2 Integrate[ f1$s, {x, 1, a}] ]


  (*  find astar  *)

rf0 =   Integrate[ D[f0$s,x]^2, {x, -1, 1}]
rf1 = 2 Integrate[ D[f1$s,x]^2, {x,  1, a}]

rfa = First[ rf0 ] + First[ rf1 ]

Plot[ rfa, {a,3.4,3.9} ]

Solve[ D[rfa,a] == 0 ]

aaa = ( N[ Solve[ D[rfa,a] == 0 ] ][[5]] )

aa = ( a /. aaa )

f0$star = First[ f0$s /. a -> aa ]
f1$star = First[ f1$s /. a -> aa ]



plt = Plot[ { f0$star Boole[ Abs[x]<1 ], f1$star Boole[ Abs[x]> 1 ] }, {x,-aa,aa} ]

f0$1 = f0$star /. x -> 1
pts=ListPlot[{{-1,f0$1},{1,f0$1},{-aa,0},{aa,0}},PlotStyle -> {Red, PointSize@Large}]

Show[ plt,pts ]

rfa /. a -> aa