(*   10-9-23   real solution   *)


fx = a + b x^2 + c x^4

cons1 = ( Integrate[ fx,{x,0,1} ] == 1/4 )
cons2 = ( ( fx /. x ->d ) == 0 )
cons3 = ( ( D[fx,x] /. x -> d ) == 0 )

abc = First[  ( Solve[ { cons1,cons2,cons3 }, {a,b,c} ] ) ]

fx$s = (  fx /. abc )

cons4 = ( Integrate[ fx$s, { x, -d, d } ] == 1 )

Solve[ cons4, d ]

dd = N[ First[ Solve[ cons4,d ] ] ]

fx$star = ( fx$s /. dd )

fx$star = ( Factor[ ( fx$s /. dd ) ] )

ddd = ( d /. dd )

Integrate[ fx$star, { x, -ddd, ddd}]
Integrate[ fx$star, { x, -ddd, ddd}]

Integrate[ fx$star, { x, 0, 1}]
Integrate[ fx$star, { x, 1, ddd}]

Integrate[ D[fx$star, x]^2, { x, -ddd, ddd}]