Find a polynomial $p$ with $p(0), p(1), p'(2)$ known

hermite-polynomialsinterpolationnumerical methods

I would like to find a polynomial $p$ so that $p(0) = 1, p(1) = 2, p'(2) = -1/2$, using Hermite interpolation, preferably with the divided differences method in the wikipedia page for Hermite interpolation.

(More generally, when we are given some functional values at some points and derivatives of varying order at other points, find the polynomial satisfying those conditions. Surely such missing-information interpolation is possible?)

Thanks in advance!

EDIT: In the slides I am reading it says

It is also possible to set up specialized Hermite interpolation
functions which do not include all functional and/or derivative values
at all nodes • There may be some missing functional or derivative
values at certain nodes • This lowers the degree of the interpolating
function

this is what I am looking for, not actual Hermite interpolation, which the kind answerers have explained to me is actually impossible

Best Answer

Applying method of differences to $p(0), p(1)$, we see that the polynomial must be of the form $ p(x) = (x+ 1) + x(x-1) \times A(x)$.

Checking the condition at $p'(2)$, we get $ - \frac{1}{2} = p'(2) = 1 + ( 4 - 1 ) \times A(x) + 2(2-1) \times A'(x) $.

So, we could have $A(x) = - \frac{1}{2}$, which yields $ - \frac{x^2}{2} + \frac{3x}{2} + 1 $.

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