Question

Let $f\left(x\right)=\{\begin{array}{cc}\text{sgn}\left(x\right),& x=0\\ a{x}^2,& x\ne 0\end{array},$ then which of the following(s) is(are) correct

• A. A point of minima exists at $x=0,$ if $a<0$
• B. A point of minima exists at $x=0,$ if $a>0$
• C. A point of maxima exists at $x=0,$ if $a<0$
• D. A point of maxima exists at $x=0,$ if $a>0$

Answer 1

Answer: (B), (C)

A point of maxima exists at $x=0,$ if $a<0$

Given : $f\left(x\right)=\{\begin{array}{cc}0,& x=0\\ a{x}^2,& x\ne 0\end{array}$, as $\text{sgn}\left(x\right)=0$ at $x=0$
Now, $f\left(0\right)=0,f(0+h)=f(0-h)=a{h}^2,h\to 0^{+}$

for point of minima at $x=0:f(0-h)>f\left(0\right)<f(0+h)$
$\Rightarrow a{h}^2>0$
$\Rightarrow a>0,$ as $h\to 0^{+}$

for point of maxima at $x=0:f(0-h)<f\left(0\right)>f(0+h)$
$\Rightarrow a{h}^2<0$
$\Rightarrow a<0,$ as $h\to 0^{+}$