1.
f(x) = a·x^3 + b·x^2 + c·x + d
2.
a, b, c, d
3.
Für jede Unbekannte eine, also 4.
4.
f(0) = 0 --> d = 0
f(2) = 7 --> 8·a + 4·b + 2·c + d = 7
f'(1) = 0 --> 3·a + 2·b + c = 0
f'(5) = 0 --> 75·a + 10·b + c = 0
5.
d einsetzen
8·a + 4·b + 2·c = 7
3·a + 2·b + c = 0
75·a + 10·b + c = 0
I - 2*II ; III - II
2·a = 7
72·a + 8·b = 0
Rückwärts lösen
2·a = 7 --> a = 3.5
72·3.5 + 8·b = 0 --> b = -31.5
3·3.5 + 2·(-31.5) + c = 0 --> c = 52.5
6.
f(x) = 3.5·x^3 - 31.5·x^2 + 52.5·x