Question

Demonstrate (by use of examples) that approximations by high order interpolating polynomials are unstable.

Answer 1

Interpolating polynomials of higher degree tend to be very oscillatory and peaked, especially near the endpoints of the interval.

which can be proved by Runge's phenomenon,

For the case of the Runge function, interpolated at equidistant points, each of the two multipliers in the upper bound for the approximation error grows to infinity with n. Although often used to explain the Runge phenomenon, the fact that the upper bound of the error goes to infinity does not necessarily imply, of course, that the error itself also diverges with n.