Question

$f\left(x\right)=\{\begin{array}{c}2a-x\text{, for}-a<x<a\\ 3x-2a\text{, for}x\ge a\end{array}$
Then which of the following is true?

• A. $f\left(x\right)$ is discontinuous at $x=a$
• B. $f\left(x\right)$ is not differentiable at $x=a$
• C. $f\left(x\right)$ is differentiable at all $x\ge a$
• D. $f\left(x\right)$ is continuous at all $x<a$

Answer 1

The correct option is B $f\left(x\right)$ is not differentiable at $x=a$

$\underset{x\to a^{-}}{lim}f\left(x\right)$

$=\underset{x\to a^{-}}{lim}2a-x$

$=2a-a$

$=a$

$=RHL$.

$\underset{x\to a^{+}}{lim}f\left(x\right)$

$=\underset{x\to a^{+}}{lim}3x-2a$

$=3a-2a$

$=a$

$=LHL$.

Now $f\left(a\right))=(3x-2a{)}_{x=a}$

$=a$.

Hence $LHL=RHL=$ value of function at 'a'.

Hence, the function is continuous at $x=a$.

Consider differentiability

$=\underset{x\to a^{-}}{lim}{f}^{\prime }\left(x\right)$

$=\underset{x\to a^{-}}{lim}=-1=LHD$

and,

$=\underset{x\to a^{+}}{lim}{f}^{\prime }\left(x\right)$

$=\underset{x\to a^{+}}{lim}3$

$=3=RHD$

Hence $LHD\ne RHD$. Thus the function is not differentiable at $x=a$.

Hence, option B is correct.