Q. $f(x) = \left\{ \begin{array}{r} 2a - x\ \ in\ –a < x < a \\ 3x - 2a\ \ in\ \ a \leq x \end{array} \right.\$. Then, which of the following is true?
A) $f(x)$ is discontinuous at $x = a$
B) $f(x)$ is not differentiable at $x = a$
C) $f(x)$ is differentiable at $x \geq a$
D) $f(x)$ is continuous at all $x < a$
Explanation
(b)
At $x = a$,
LHL$= \lim_{x \rightarrow a^{-}}\ f(x) = \lim_{x \rightarrow a}2a - x = a$
And RHL$= \lim_{x \rightarrow a^{+}}\ f(x) = \lim_{x \rightarrow a}3x - 2a = a$
And $f(a) = 3(a) - 2a = a$
$\therefore$ LHL=RHL$= f(a)$
Hence, it is continuous at $x = a$
Again, at $x = a$
LHD$= \lim_{h \rightarrow 0}\ \frac{f(a - h) - f(a)}{- h}$
$= \lim_{h \rightarrow 0}\ \frac{2a - (a - h) - a}{- h} = - 1$
and RHD$= \lim_{h \rightarrow 0}\ \frac{f(a + h) - f(a)}{h}$
$= \lim_{h \rightarrow 0}\ \frac{3(a + h) - 2a - a}{h} = 3$
$\therefore$ LHD$\neq$RHD
Hence, it is not differentiable at $x = a$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.