Question

Let $f:R\to R$ be defined by
$f\left(x\right)=\{\begin{array}{c}k-2x,ifx\le -1\\ 2x+3,ifx>-1\end{array}$
If $f$ has a local minimum at $x=-1$, then a possible value of $k$ is

• A. $-1/2$
• B. $-1$
• C. $1$
• D. $0$

Answer 1

The correct option is B $-1$

If $f\left(x\right)$ has a local minimum at $x=-1$, then
${lim}_{x\to -1^{+}}f\left(x\right)={lim}_{x\to {-1}^{-}}f\left(x\right)$
$\Rightarrow {lim}_{x\to -1^{+}}2x+3={lim}_{x\to {-1}^{-}}k-2x$
$\Rightarrow -2+3=k+2\Rightarrow k=-1$