I don't know anything about that Hermite-something. But I played with your question using what little I know about points, gradients and characteristics of some known equations/functions. And I found an equation/function that you're looking for.

Graphing those ordered pairs on the same x,y plane, I saw that the graph of the unknown function is rising steeply at point (-2,-57), then it level off at (0,3), then goes down after (0,3), and then rises steeply again at (3,3).

That looks like a vertical S-curve. A cubic curve in x.

So I played with:

f(x) = Ax^3 +Bx^2 +Cx +D -----(1)

So,

f'(x) = 3Ax^2 +2Bx +C -----------(2)

At (x=0, f'(x)=0), in (2):

0 = 3A(0) +2B(0) +C

C = 0 --------------------------***

So, (2) becomes

f'(x) = 3Ax^2 +2Bx ------------(2a)

At (x=3, f'(x)=27), in (2a):

27 = 3A(3^2) +2B(3)

9 = 9A +2B --------------------(3)

At (x=-2, f'(x)=72), in (2a):

72 = 3A((-2)^2) +2B(-2)

72 = 12A -4B

18 = 3A -B

B = 3A -18

Plug that into (3),

9 = 9A +2(3A -18)

9 = 9A +6A -36

A = 45/15 = 3 ---------------------------***

So, (2a) becomes f'(x) = 9x^2 +2Bx -----------(2b)

Back to (x=3, f'(x)=27), in (2b):

27 = 9(3^2) +2B(3)

27 = 81 +6B

B = (27-81)/6 = -9 ---------------------***

So (1) becomes f(x) = 3x^3 -9x^2 +D -----(1a)

At (x=0, f(x)=3), in (1a):

3 = 3(0) -9(0) +D

D = 3 ---------------------------------***

So (1a) becomes f(x) = 3x^3 -9x^2 +3 .

And, f(x) = 3x^3 -9x^2 +3 is the function you are looking for.

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I checked that on all the given ordered pairs and it checked allright.