Question

If f' (x) = x + b, f(1) = 5, f(2) = 13, find f(x)

Answer 1

$$\begin{gathered} f\text{'}\left(x\right)=x+b,f\left(1\right)=5,f\left(2\right)=13 \\ f\text{'}\left(x\right)=x+b \\ \int f\text{'}\left(x\right)dx=\int \left(x+b\right)dx \\ f\left(x\right)=\frac{x^2}{2}+bx+\mathrm{C}....\left(i\right) \\ f\left(1\right)=5,f\left(2\right)=13\left(\mathrm{Given}\right) \\ \mathrm{Puting}x=1\mathrm{in}\left(i\right) \\ f\left(1\right)=\frac{1^2}{2}+b1+\mathrm{C} \\ 5=\frac{1}{2}+b+\mathrm{C}...\left(ii\right) \\ \mathrm{Puting}x=2\mathrm{in}\left(i\right) \\ f\left(2\right)=\frac{2^2}{2}+b2+\mathrm{C} \\ 13=\frac{4}{2}+2b+\mathrm{C} \\ 13=2+2b+\mathrm{C}...\left(iii\right) \\ \mathrm{Solving}\left(ii\right)\mathrm{and}\left(iii\right)\mathrm{we}\mathrm{get}, \\ b=\frac{13}{2}\mathrm{and}C=-2 \\ \mathrm{Thus},f\left(x\right)=\frac{x^2}{2}+\frac{13}{2}x-2 \end{gathered}$$