Question

Find f. f ''(x) = −2 + 36x − 12x2, f(0) = 8, f '(0) = 18 f(x) =

Answer 1

Integrate f ''(x) = −2 + 36x − 12x2 with respect to x:

f '(x) = -2x + (36/2)x^2 - (12/3)x^3 + c.  Find c by letting x = 0 and using f(0)=8.

Then f '(0) = -2x + 18x^2 - 4x^3 + c = 18 (which was given).

Then             -0      + 0      - 0 + c = 18, so c = 18 and 

f '(x) = -2x + 18x^2 - 4x^3 + 18.

Go through the same integration process to find f(x).