Basis functions for polynomial

[Somefantastik]

Homework Statement

For I = [a,b], define: P₃(I) = {v: v is a polynomial of degree ≤ 3 on I, i.e., v has the form v(x) = a₃x³ + a₂x² + a₁x + a₀}. How to show v is uniquely determined by v(a), v'(a), v(b), v'(b).

Homework Equations

The Attempt at a Solution

I'm not exactly sure what I'm being asked to do here. I don't need the problem solved, just a nudge in the right direction.

 

[epenguin]

I guess I = [a,b] just stands for an interval of real numbers?

I also guess that you are allowed to assume that multiplication and addition of given real numbers always produces a unique real number?

Then your four given coefficients a_i uniquely determine P.

So it seems you are being asked to prove that v(a), v'(a) etc. uniquely determine the a_i .