/*  IQR Oversmoothing PDF  */

f0 = b - c x^2
f1 = d * (a^2-x^2)

   /* constraints */
   
cons1 = ( ( f0 /. x -> 1 ) == ( f1 /. x -> 1 ) )

cons2 = ( Integrate[ f0,{x,0,1} ] == 1/4 )

cons3 = ( Integrate[ f1,{x,1,a} ] == 1/4 )


bcd = ( Solve[ {cons1,cons2,cons3}, {b,c,d} ] )


f00 = (  f0 /. bcd  )
f11 = (  f1 /. bcd  )

rfp0 = 2 Integrate[ D[f00,x]^2,{x,0,1} ]
rfp1 = 2 Integrate[ D[f11,x]^2,{x,1,a} ]

rfp = rfp0 + rfp1

Plot[rfp,{a,2.8,3.6}]

astar = N[ Last[Solve[ D[rfp, a] == 0, a]] ]

f000 = f00 /. astar
f111 = f11 /. astar

aa = ( a /. astar )

Plot[ { f000 Boole[ Abs[x]<1 ], f111 Boole[ Abs[x]>1]},{x,-aa,aa} ]

2 Integrate[ f000, {x,0,1} ]
2 Integrate[ f111, {x,1,aa} ]

ans = ( rfp /. a -> aa )