Question

Let $f\left(x\right)=\{\begin{array}{cc}{x}^2,& x\ge 0\\ ax,& x<0\end{array}$
The set of real values of $a$ such that $f\left(x\right)$ will have local minima at $x=0,$ is:

• A. $a\in R$
• B. $a\in R^{-}$
• C. $a\in \varphi$
• D. $a\in R^{+}$

Answer 1

The correct option is B $a\in R^{-}$

Given : $f\left(x\right)=\{\begin{array}{cc}{x}^2,& x\ge 0\\ ax,& x<0\end{array}$
$\Rightarrow f\left(0\right)=0,f(0+h)=h^2>0$ and $f(0-h)=-ah,h\to 0^{+}$
For point of minima at $x=0:f(0-h)>f\left(0\right)<f(0+h)$ as $h\to 0^{+}$
$\Rightarrow -ah>0$
$\Rightarrow a<0$ as $h\to 0^{+}$