Consider the function f(x) whose second derivative is f"(x)=4x+10sin(x). If f(0)=3 and f'(0)=2, what is f(4)?

Question

Consider the function f(x) whose second derivative is f"(x)=4x+10sin(x). If f(0)=3 and f'(0)=2, what is f(4)?

Answer 1

f'(x) = ∫(4x + 10sinx)dx = 2x² - 10cosx + C; f'(0) = - 10 + C = 2; C = 12 and f'(x) = 2x² - 10cosx + 12;

f(x) = ∫(2x² - 10cosx + 12)dx = 2x³/3 - 10sinx + 12x + M; f(0) = M = 3; f(x) = 2x³/3 - 10sinx + 12x + 3;

f(4) = 128/3 - 10sin4 + 48 + 3 = 281/3 - 10sin4.

Answer 2

f ''(x)=4x+10sin(x)

f '(x)=2x²-10cos(x)+C₁

f '(0) =-10+C₁=2

C₁=12

f(x)=2x³/3-10sin(x)+12x+C₂

f(0)=C₂=3

f(x)=2x³/3-10sin(x)+12x+3

Now, set x to 4 to find f(4)