Find f. f ''(x) = −2 + 36x − 12x^2, f(0) = 3, f '(0) = 14

Question

Find f. f ''(x) = −2 + 36x − 12x^2, f(0) = 3, f '(0) = 14

Answer 1

f"(x) = -2 +36x - 12x^2 integrate to get

f'(x) =-2x +18x^2 -4x^3 +c

f'(0) =14 = c

f'(x) =-2x +18x^2 -4x^3 +14 integrate to get

f(x) =-x^2 +6x^3 -x^4 +14x + k

f(0) =3 = k

f(x) = -x^4 +6x^3 -x^2 +14x +3

Answer 2

This is an antiderivative initial value problem. Start by finding f '(x) and use f '(0) = 14 to find +C. Then use the same process to find f(x) using f(0)=3 to find the second C.

f '(x) = -2x+18x^2-4x^3+C

f '(0)=14 --> C=14

f(x) = -x^2+6x^3-x^4+14x+D

f (0) = 3 --> D = 3

Hope this helps!