Question

If $f\left(x\right)=\{\begin{array}{cc}|x-2|+a,& x\le 2\\ 3x-1,& x>2\end{array}$ has a local minimum at $x=2$, then $a=$

• A. $5$
• B. $4$
• C. $1$
• D. $2$

Answer 1

The correct option is A $5$

$f\left(x\right)=\{\begin{array}{cc}2+a-x,& x\le 2\\ 3x-1,& x>2\end{array}$
Left hand derivative of $f\left(x\right)$ at $x=2$ is $-1<0$ and RHD $=3>0$. .
$\therefore f\left(x\right)$ is decreasing in $(-\infty ,2]$ and increasing in $(2,\infty )$.
$\Rightarrow$ minimum value of f is $=(3\times 2-1)=2+a-2$
$\Rightarrow a=5$