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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions1 Mark
Question
Function f(x) = |x| - |x - 1| is monotonically increasing when:
  1. x < 0
  2. x > 1
  3. x < 1
  4. 0 < x < 1
✓

Answer

  1. 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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