GATE Overflow Test Series | Calculus | Test 1 | Question: 19

Question

Let $f(x)$ be the function defined as $$f(x)= \left\{
\begin{array}{ll}
\mid x\mid  & \text{for } 0 < x \leq 2 \\
1 & \text{for } x= 0
\end{array}
\right. $$
At $x = 0, f$ has:

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

Answer 1

At the point $x = 0$ function is not continuous as limit value not equal to $f(0)$. So, it is a critical point and can be either a local maxima or local minima. Since, $x=0$ is a boundary point and $\displaystyle \lim_{x \to 0} f(x) = 0,$ and $f(0) = 1,$ at $x=0, f(x)$ is having a local maximum.

Answer 2

Given that, $f(x)= \left\{
\begin{array}{ll}
\mid x\mid  & \text{for } 0 < x \leq 2 \\
1 & \text{for } x= 0
\end{array}
\right. $

$\therefore x = 0,$ has a local maximum.

References:

• https://ncert.nic.in/ncerts/l/leep206.pdf
• https://www3.nd.edu/~apilking/Math10550/Lectures/15.%20Maxima%20and%20Minima.pdf