find f if f''(x)=

Question

Find f if f′′(x)=sin x + cos x, f′(0)=3, and f(0)=2.

 

f(x)=_____

Answer 1

f' = -cos x + sin x + C (because its derivative gives you exactly the stated f") and f'(0) = 3 = -1 + 0 + C and thus C=4.

 

f = ∫ [ -cos x + sin x + 4]dx = 4x - sin x - cos x + K (check by differentiation, to get f' ).

 

Using the f(0) = 2 leads me to the conclusion that K = 3.

 

Answer: f(x) = 4x - sin x - cos x + 3