e[b_] := b^2 (1 - b^2/3)/(1 - 2 b^2/3 + b^4)
dE = e'[b]

Solve[dE == 0]

u[x_] := A ( 1 - (b x/a)^2)
norm := Integrate[ u[x] u[x], {x, -a, a}]
eE = -(ℏ^2/2 m) Integrate[ u[x] u''[x], {x, -a, a}]

ebE = eE/norm

deEb = D[ebE, b]

Solve[deEb == 0]

{{b->0},{b->-(Sqrt[3 (5-2 (Sqrt[5]))])},{b->(Sqrt[3 (5-2 (Sqrt[5]))])},{b->-(Sqrt[3 (5+2 (Sqrt[5]))])},{b->(Sqrt[3 (5+2 (Sqrt[5]))])},{m->0},{ℏ->0}} // N

Pi/2 // N

(* 1.258 fairly close to exact second order taylor series expansion of psi ~ Cos[ Pi x/ 2a]  which gives psi ~ 1 - (Pi x/2 a)^2 ;  ie: b = Pi/2 *)