We can find our polynomial interpolation by writing g as a general polynomial of the given degree,expressing each of our conditions as a relation among g's coefficients, and solving the resulting system of linear equations to determine these coefficients. This can be done similarly for eitherproblem; the only difference is that in one case some of the equations come from derivatives of g.
We start this process for our first problem where the derivatives of g must be continuous at b and c.
g(x) = g0 +g1 x+g2 x2+g3 x3.
Our four conditions become:
g0+ g1 b + g2 b2 + g3 b3 = f(b)
g1 + 2g2 b + 3g3 b2 = f '(b)
g1 + 2g2 c + 3g3 b2 = f '(c)
g0 + g1 c + g2c2 + g3 c3 = f(c)
In the second problem there are k equations, each like the first and fourth here going up to degree k - 1.
g0 + g1 xj + g2 xj2 +...+ g k -1 xjk-1 = f(xj )
The pedestrian approach is to write down all these equations and use algebraic manipulations to solve them.