$\text{ If }f\left(x\right)=\left\{\begin{array}{ll}x& ,x\le 1\\ x^2+bx+c& x>1\end{array}\text{ then find value of }b\text{ and }c.\text{ If }f\left(x\right)\text{ is differentiable at }x=1\right.$

$f^1\left(x\right)=\left\{\begin{array}{l}1\text{ , }x<1\\ 2x+b\text{ , }x>1\end{array}\right.$

$f\left(x\right)\text{ is differentiable at }x=1$

$\therefore f\text{ is continuous of }x=1$

$\begin{array}{l}\underset{x\to 1+}{Lt}f\left(x\right)={Lt}_{x\to 1-}f\left(x\right)\\ 1+b+c=1\Rightarrow b+c=0\\ \text{ Also }{f}^1\left(1^{+}\right)=f^1\left(1^{-}\right)\\ Lt{f}^1\left(x\right)=Lt{f}^1\left(x\right)\\ 2+b=1\\ \Rightarrow b=-1\\ \therefore c=1\end{array}$