Find f. f=______

Question

f ''(x) = −2 + 12x − 12x²,    f(0) = 5,    f '(0) = 12

Answer 1

integrating:

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

f'(0) = 12 ---> C = 12

so f'(x) = -4x^2 + 6x^2 -2x + 12

integrating again:

f(x) = -(4/3)x^3 + 2x^3 - x^2 + 12x + C

f(0) = 5 ---> C = 5

f(x) = -(4/3)x^3 + 2x^3 - x^2 + 12x + 5

Answer 2

Hey! Lets solve this shall we ?

So, we must find the function f by computing about two anti-derivatives. Then, by using the anti-derivative of the Power Rule, we obtain

f '' (x) = -2 + 12x - 12x²

f ' (x) = -2x + (12)(x²/2) - (12)(x³/3).........Then we simplify it to become

f ' (x) = -2x + 6x² - 4x³..........Then we finalize by computing another anti-derivative such that

f(x) = (-2)(x²/2) + (6)(x³/3) - (4)(x⁴/4).........Then simplify!

I hope this helped!