Question

Statement-1: $f\left(x\right)=|\left[x\right]x|forx\in [-1,2],$ where [.] represents greatest integer function, is not differentiable from the left at $x=2$
Statement-2: A discontinuous function is non differentiable.

• A. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
• B. Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
• C. Statement-1 is True, Statement-2 is False
• D. Statement-1 is False, Statement-2 is True

Answer 1

Answer: (A)

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

$f\left(x\right)=\mid \left[x\right]x\mid$ for $xϵ[-1,2]$
L.H.D. at $x=2$
$=\underset{h\to 0}{lim}\frac{f(2-h)-f\left(2\right)}{-h}$
$=\underset{h\to 0}{lim}\frac{\mid [2-h](2-h)\mid -4}{-h}$
$=\underset{h\to 0}{lim}\frac{\mid 2-h\mid -4}{-h}$
$=\underset{h\to 0}{lim}\frac{-2-h}{-h}$
$=-\infty$
Since, the limit is not a finite number, the function is not differentiable from left at $x=2$.
Statement-I is true.
Statement-2:
A discontinuous function is non-differentiable.
Statement-2 is also true.
Check for continuity at $x=2$.
$\underset{x\to 2^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}\mid [2-h](2-h)\mid$
$=\underset{h\to 0}{lim}\mid 2-h\mid$
$=2$
$\underset{x\to 2^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}\mid [2+h](2+h)\mid$
$=\underset{h\to 0}{lim}\mid 4+2h\mid$
$=4$
$\underset{x\to 2}{lim}$ does not exist.
Hence, Statement 2 is a correct explanation of statement 1.