Question

Which one of the following is not true?

• A. $f(x)=x|x|$ is differentiable in $(-1,1)$
• B. $g(x)=\sqrt{|x|}$ is differentiable in $(4,5)$
• C. $h(x)=|x-2|+|x+3|$ is differentiable in $(3,2)$
• D. $k(x)=|x+1|+|x-6|$ is differentiable in $(-1,6)$
• E. $t(x)=x+[x]$, where $[x]$ is greatest integer function, is differentiable at $x=0$

Hint:

Differentiability requires continuity first—check for jumps or corners.

Answer 1

Answer: (E)

Concept:
• Absolute value functions are non-differentiable at points where expression inside becomes zero
• Greatest integer function is discontinuous at integers

Step 1: Check option (E)
\[ t(x)=x+[x] \] At $x=0$, $[x]$ has jump discontinuity $\Rightarrow$ not continuous $\Rightarrow$ not differentiable

Step 2: Check others briefly
(A) Smooth in interval $\Rightarrow$ differentiable
(B) Away from 0 $\Rightarrow$ differentiable
(C) No corner in interval $\Rightarrow$ differentiable
(D) Corners at $-1,6$ but excluded $\Rightarrow$ differentiable Final Conclusion:
Option (E) is not true.