Question

If $f^{\prime }\left(x\right)=x+b,f\left(1\right)=5,f\left(2\right)=13$, find $f\left(x\right)$.

Answer 1

$f^{\prime }\left(x\right)=x+b$

$\therefore f\left(x\right)=\int x+b$

$\therefore f\left(x\right)=\frac{x^2}{2}=bx=c$

$\therefore f\left(1\right)=5=\frac{1}{2}+b+c$

$\therefore b+c=\frac{9}{2}-\left(1\right)$

$\therefore f\left(2\right)=2+2b+c=13$

$\therefore b+c=11$

$\therefore b=11-\frac{9}{2}$

$\therefore b=\frac{11}{2}$

$\therefore c=-2$

$f\left(x\right)=\frac{x^2}{2}+\frac{13x}{2}+-2$