Question

Let $$\begin{gathered} f\left(x\right)=\left\{\left|x\right|for0<\left|x\right|\le 2 \\ 1forx=0\right\} \end{gathered}$$ then $x=0$ has:

• A. a local maximum
• B. no local maximum
• C. a local minimum
• D. no extremum

Answer 1

The correct option is D

no extremum

Explanation for the correct option:

According to the given function, if we plot $f\left(x\right)=\left|x\right|$, the graph looks like this.

So, it is evident that there is a sharp point at $x=0$ which makes the function non-differentiable at that point.

Hence, $f\left(x\right)$ has no extremum at $x=0$.

Therefore, the correct answer is Option (D).