Polynomial functions and their roots

problem
Patterns of inflection

Age

16 to 18

Challenge level

Find the relationship between the locations of points of inflection, maxima and minima of functions.
problem
Polynomial interpolation

Age

16 to 18

Challenge level

Can you fit polynomials through these points?
problem
Interpolating polynomials

Age

16 to 18

Challenge level

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.