Calculus 1 help?

Question

Consider the function f(x)  whose second derivative is f′′(x)=8x+5sin(x). If f(0)=2  and f′(0)=2  , what is f(x) ?

Answer 1

ANTIDIFFERENTIATING f" gives f'(x) = 4x² -5cosx + C

f'(0) = 2 = 4(2²) - 5 cos 0 + C

2 = 16 - 5 + C  so C = -9.

 

Antidifferentiating f' gives (4/3)x³ - 5 sinx - 9x + K

f)0) = 2 = K

 

Final answer: f(x) = (4/3)x³ - 5sinx - 9x +2