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

Question

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

x < 0
x > 1
x < 1
0 < x < 1

✓

Answer

0 < x < 1

Solution:
f(x) = |x| - |x - 1|
Case I:
Let x < 0
If x < 0, then |x| = -x
⇒ |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 II:
Let x < 0 < 1
Here,
|x| = x
⇒ |x - 1| = -(x - 1)
Now,
f(x) = |x| - |x - 1|
= x + x -1
= 2x - 1

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