Let $f\left(x\right)=\left\{\begin{array}{cc}|x-1|+a,& x\le 1\\ 2x+3,& x>1\end{array}\right.$If$f \left(x\right)$ has local minimum at$x = 1$ and $a \ge 5$ then the value of a is

$f\left(x\right)=\left\{\begin{array}{cc}1-x+a,& x\le 1\\ 2x+3,& x>1\end{array}\right.$

If$f \left(x\right)$has local minimum at $x = 1,$ then $f \left(1\right) \le$

$f(1+h),h>0\text{ i.e. }a\le 2(1+h)+3$

Hence $a\le 5.\text{ So }a=5$