Question

Consider the function f(x) whose second derivative is f′′(x)=8x+5sin(x). If f(0)=2 and f′(0)=3, what is f(x)?

Answer 1

f" = 8x + 5sinx

f' = 4x^2 - 5cosx+C
f'(0) = -5+C = 3 so C = 8
f = 4/3 x^3 - 5sinx + 8x + C
f(0) = 8+C = 2 so C = -6
f(x) = 4/3 x^3 - 5sinx + 8x - 6