help: talyor series, functional extrema

Hi,

I would really appreciate any help. :)

Let's say I'm looking for a *function* Y that minimizes an equation.

If I approximate Y with a polynomial series of order N: Yp =
sum_over_N(a_i * x^i)
take its deriviatve, set it to zero, and solve for a_i and find that
all the a_i have a single solution, i.e. linear equations in a_i, does
this tell me anything about Y.  Does it limit the space of possible Y?

Thanks for any help! 

james