For a set of control points $\{(0,0),(2,4),(3,5),(4,3),(7,2)\}$ build the Hermite interpolation polynomial $ H(x),$ if all derivatives up to the 4th order at the center point are additionally known: $H'(3) = 1, H''(3) = -1, H'''(3) = 0, H''''(3) = -2. $
My attempt: enter image description here
Try tu use the triangular diagram similar to the Newton interpolation polynomial and use divided differences.
What about $H^{v}(3)$?
Is this hint it enough?
The start of the diagram is here. The rest of computation is easy to make a mistake. :)
In the 3rd column we have 1 because this is $H'(3)$. In the 4th column we have $-1/2$ because this is $f''(3)/2!$. In the next column there will be $0$ as $H'''(3)/3!$, in the next one $h''''(3)/4!=2/24=1/12.$ The rest of the divided differences is computed by the standard way.