Function f(x) = |x| - |x - 1| is monotonically increasing when : - Vidyadip

MCQ

Function $f(x) = |x| - |x - 1|$ is monotonically increasing when :

A
$x < 0$
B
$x > 1$
C
$x < 1$
✓
$0 < x < 1$

✓

Answer

Correct option: D.

$0 < x < 1$

$f(x) = |x| - |x - 1|$
Case $\text{I} :$
Let $x < 0$
If $x < 0,$ then $|x| = -x$
$\Rightarrow |x - 1| = -(x - 1)$
Now,
$f(x) = |x| - |x - 1|$
$= -x - (-x + 1)$
$= -x + x - 1$
$= -1$
$f'(x) = 0$
So, $f(x)$ is not monotonically increasing when $x < 0.$
Case $\text{II} :$
Let $x < 0 < 1$
Here,
$|x| = x$
$\Rightarrow |x - 1| = -(x - 1)$
Now,
$f(x) = |x| - |x - 1|$
$= x + x -1$
$= 2x - 1$

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