Problem
Ouestion 3: (1 mark) The function \( f(x) \) whose tangent has the slope \( 6 x^{2}-2 x+6 \) and whose graph passes through the point \( (1,-3) \) is A) \( f(x)=2 x^{3}+x^{2}+6 x-10 \) B) \( f(x)=2 x^{3}-x^{2}-6 x-10 \) C) \( f(x)=2 x^{3}+x^{2}-6 x-10 \) D) \( f(x)=2 x^{3}-x^{2}+6 x-10 \)
Solution
Step 1 ：\(f'(x)=6x^2-2x+6\)
Step 2 ：\(f(x)=\int (6x^2-2x+6) dx=2x^3-x^2+6x+C\)
Step 3 ：\(f(1)=-3\: \Rightarrow 2(1)^3-(1)^2+6(1)+C=-3 \: \Rightarrow C=-10\)
Step 4 ：\(f(x)=2x^3-x^2+6x-10\)
