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'(x) = 4x² - 5 cos x + C

f(x) = 4x³/3 - 5 sin x + Cx + D

Now use the 2 initial conditions to evaluate C and D

Answer 2

First we find the integral of f''(x)

f '(x)=∫ 8x + 5 sin(x) dx + C

f ' (x) = 4x² - 5 cos (x) + C

it is given that f ' (0) = 3, so C=3 so

f ' (x) = 4x² - 5 cos (x) + 3

now we find the integral of f ' (x)

f (x) =∫ 4x² - 5 cos (x) + 3 dx +C

f(x)= (4/3)x³ - 5 sin(x) +3x + C

it is given that f (0) = 2, so C=2 so our final answer is

f(x)= (4/3)x³ - 5 sin(x) +3x + 2