




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.
Set
g(x) = g_{0 }+g_{1} x+g_{2 }x^{2}+g_{3 }x^{3}.
Our four conditions become:
g_{0}+ g_{1 }b + g_{2 }b^{2 }+ g_{3 }b^{3 }= f(b)
g_{1 }+ 2g_{2} b + 3g_{3} b^{2 }= f '(b)
g_{1 }+ 2g_{2} c + 3g_{3} b^{2 }= f '(c)
g_{0 }+ g_{1 }c + g_{2}c^{2 }+ g_{3} c^{3 }= f(c)
In the second problem there are k equations, each like the first and fourth here going up to degree k  1.
g_{0 }+ g_{1} x_{j }+ g_{2} x_{j}^{2 }+...+ g_{ k 1} x_{j}^{k1 }= f(x_{j })
The pedestrian approach is to write down all these equations and use algebraic manipulations to solve them.