D-admissible directions and extermal points

[braindead101]

Let X = C[a,b], J(y) = integ(a to b) sin^3(x) + y(x)^2 dx and D={yEX; integ(a to b) y(x)dx = 1}
(a) what are the D-admissible directions for J?
(b) Find all possible (local) external points for J on D?

so far i have:
(let e be epsilon)
lim e->0 J(y+ev) - J(y) / e
= lim e->0 integ (a to b) [ sin^3(x) + (y+ev)^2(x)dx - J(y) ] / e
= lim e->0 integ (a to b) [2ey(x)v(x) + e^2v(x)^2) dx ] / e
= 2 integ(a to b) y(x)v(x) dx
A similar example was done in class, so i just copied it for this question.
G(y) = integ(a to b) y(x) dx = 1
gateaux G(y;v) = integ (a to b) y(x)v(x) dx

is this right so far? how do i go about answering (a) and (b).

 

[braindead101]

i was wondering if the D-admissible directions for J is just
y+Ev E D
so,
integ (a-b) (y+Ev)(x) dx
= 1+ E integ(a-b) v(x) dx

can i simplify any further?