Question

Find f.
f ''(x) = 20x3 + 12x2 + 10,    f(0) = 9,    f(1) = 12

Answer 1

integrating once to get
f'(x) = 5x^4 + 4x^3 + 10x + c
check by taking the derivative,

integrate once more to get
f(x) = x^5 + x^4 + 5x^2 + cx + k

now use the two given points:
for f(0) =9
9 = 0+0+0+0+k , so k = 9
for (1,12)
12 = 1+1+5+c+9
c = ...

f(x) = ....