Science:Math Exam Resources/Courses/MATH100/December 2019/Question 4 (a)/Solution 1

Integrate f"(x) twice. Integrating f"(x) once we obtain:

$\int f''(x)dx=\int \cos(x)+2dx=\sin(x)+2x+C$

for some constant C. Therefore, by the Fundamental Theorem of Calculus, f'(x) = sin(x) + 2x +C₁ . Integrating this expression again yields

$f(x)=-\cos(x)+x^{2}+C_{1}x+C_{2}.$

Now, f(0) = -cos(0) + C₂= -1+C₂=5 and we conclude that C₂=6. On the other hand, f'(0) = sin(0) +C₁=5 , which yields C₁=5.

In conclusion, the desired function is $f(x)=-\cos(x)+x^{2}+5x+6$.