Question

The set of all real value(s) of $a$ for which function $f\left(x\right)=\{\begin{array}{cc}2x,& -1\le x\le 0\\ -3x+a,& 0<x\le 2\end{array}$ has a local maxima at $x=0,$ is

• A. $(-\infty ,0]$
• B. $(-\infty ,2)$
• C. $(0,\infty )$
• D. $\varphi$

Answer 1

The correct option is A $(-\infty ,0]$

Given : $f\left(x\right)=\{\begin{array}{cc}2x,& -1\le x\le 0\\ -3x+a,& 0<x\le 2\end{array}$
For local maxima at $x=0:$
$f(0-h)<f\left(0\right)>f(0+h)$ as $h\to 0^{+}$
$\Rightarrow 2(0-h)<0$ and $-3(0+h)+a<0$ as $h\to 0^{+}$
$\Rightarrow -2h<0$ and $a<3h$ as $h\to 0^{+}$
$\Rightarrow a\in R$ and $a\le 0$
$\therefore a\in (-\infty ,0]$