Question

f''(x)=4x-1 f'(2)=-2;f(1)=3

Answer 1

$\Large\boxed{\begin{array}{l}\displaystyle\sf f'(x)=\int f''(x)~dx\\\displaystyle\sf f'(x)=\int (4x-1)dx\\\sf f'(x)=2x^2-x+c\\\sf f'(2)=2\cdot2^2-2+c\\\sf -2=8-2+c\\\sf c+6=-2\\\sf c=-2-6\\\sf c=-8\\\sf f'(x)=2x^2-x-8.\end{array}}$

$\Large\boxed{\begin{array}{l}\displaystyle\sf f(x)=\int f'(x)\,dx\\\displaystyle\sf f(x)=\int(2x^2-x-8)dx\\\\\sf f(x)=\dfrac{2}{3}x^3-\dfrac{1}{2}x^2-8x+c_2\\\\\sf f(1)=\dfrac{2}{3}\cdot 1^3-\dfrac{1}{2}\cdot 1^2-8\cdot1+c_2\\\\\sf 3=\dfrac{2}{3}-\dfrac{1}{2}-8+c_2\bullet(6)\\\\\sf 18=4-3-48+6c_2\\\sf 6c_2=18-4+3+48\\\sf 6c_2=65\\\\\sf c_2=\dfrac{65}{6}\end{array}}$

$\Large\boxed{\begin{array}{l}\huge\boxed{\boxed{\boxed{\boxed{\sf f(x)=\dfrac{2}{3}x^3-\dfrac{1}{2}x^2-8x+\dfrac{65}{6}}}}}\end{array}}$