question_answer

Let \[f(x)=\left\{ \begin{matrix}    |x|,\,\,for\,\,0<|x|\le 2  \\    1,\,\,\,\,\,for\,\,\,\,x=0  \\ \end{matrix} \right.\], then at \[x=0,\] \[f\] has

A)  a local maximum

B)  a local minimum

C)  no local extremum

D)  no local maximum

Correct Answer: C

Solution :

Given that, \[f(x)=\left\{ \begin{matrix}    |x|,\,\,\,\,for\,\,\,0<|x|\le 2  \\    1,\,\,\,\,\,\,\,\,for\,\,\,\,x=0  \\ \end{matrix} \right.\] It is clear from the graph that \[f(x)\] is not continuous and differentiable at \[x=0\]. Hence, it has no local extremum.