Question

Let $f\left(x\right)=\{\begin{array}{cc}2a-x,& if-a<x<a\\ 3x-2a,& ifa\le x\end{array}$. 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 $x\ge a$
• D. $f\left(x\right)$ is continuous at all $x<a$

Answer 1

Answer: (B)

$f\left(x\right)$ is not differentiable at $x=a$

Given, $f\left(x\right)=\{\begin{array}{cc}2a-x,& if-a<x<a\\ 3x-2a,& ifa\le x\end{array}$
At $x=a$,
$LHL=\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a}{lim}(2a-x)$
$=2a-a=aRHL$
$=\underset{x\to a^{+}}{lim}f\left(x\right)=\underset{x\to a}{lim}(3x-2a)$
$=3\left(a\right)-2a=a$
and $f\left(a\right)=3\left(a\right)-2a=a$
$\therefore LHL=RHL=f\left(a\right)$
Hence, it is continuous at $x=a$
Again, at $x=a$
$LHD=\underset{h\to 0}{lim}\frac{f(a-h)-f\left(a\right)}{-h}$
$=\underset{h\to 0}{lim}=\frac{2a-(a-h)-a}{-h}=-1$
and $RHD=\underset{h\to 0}{lim}\frac{f(a+h)-f\left(a\right)}{h}$
$=\underset{h\to 0}{lim}\frac{3(a+h)-2a-a}{h}=3$
$\therefore LHD\ne RHD$
Hence, it is not differentiable at $x=a$.