Looks good to me!
in order to find the vertex of this problem:
y=1/2x^2+1/3x
you set it up in this formula:
y=m(x-h)^2+k
because (h,k) is the vertex.
is that correct so far?
if so then use this:
[ (-b/2a),f(-b/2a) ]
which gets (1/3, 1/18)
is this correct?