In our first problem the nature of the solution depends on what we mean by a smooth interpolation.

If we only want the interpolation g to agree with f at its endpoints, b and c, then we have two conditions:

f(b) = g(b), f(c) = g(c)

and we can find a straight line that obeys them;

g(x) = f(c)(x - b) / (c - b) + f(b)(x - c)/(b - c).

Such an interpolation will usually not be differentiable at b or c; it is therefore customary to require more of g, and in particular that it have the same derivative as f at b and c.

g'(b) = f '(b), g'(c) = f '(c).

We then have four conditions on g, and in general need a cubic to meet them. We could insist that the second derivatives of g match those of f at b and c, and seek a fifth degree curve that accomplishes all this, and so on.