Antiderivative help!

Question

given 

 

f''(x)=6x+2

and f'(-3)=2 and f(-3)=-5

 

find f'(x)=?

and find f(4)=?

Answer 1

f'(x) is the antiderivative of f''(x).

 

f'(x) = 3x² + 2x + C

 

 

f(x) is the antiderivative of f'(x).

 

f(x) = x³ + x² + Cx + D

 

where C and D are arbitrary constants.    Use the initial value conditions to solve for C and D.

 

 

From the f'(x) condition,

 

3(-3)² + 2(-3) + C = 2

 

3(9) - 6 + C = 2

 

               C = 2 - 27 + 6

 

               C = -19

 

 

From f(x) condition,

 

(-3)³ + (-3)² - 19(-3) + D = -5

 

-27 + 9 + 57 + D = -5

 

-27 + 66 + D = -5

 

39 + D = -5

 

        D = -44

 

 

f'(x) = 3x² + 2x - 19

 

f(x) = x³ + x² - 19x - 44

 

 

You can find f(4).