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Increasing and Decreasing Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIncreasing and Decreasing Functions1 Mark
Question
The function $\text{f}(\text{x})=\frac{\text{x}}{1+|\text{x}|}$ is:
  1. Stritcly increasing.
  2. Stritcly decreasing.
  3. Neither increasing nor decreasing.
  4. None of these.
✓

Answer

  1. Stritcly increasing.
Solution:
$\text{f}(\text{x})=\frac{\text{x}}{1+|\text{x}|}$
Case I:
When x > 0, |x| = x
$\text{f}(\text{x})=\frac{\text{x}}{1+|\text{x}|}$
$=\frac{\text{x}}{1+\text{x}}$
$\Rightarrow\text{f}'(\text{x})=\frac{(1+\text{x})1-\text{x}(1)}{(1+\text{x})^2}$
$=\frac{1}{(1+\text{x})^2}>0,\forall\ \text{x}\in\text{R}$
So, f(x) is strictly increasing when x > 0.
Case II:
When x < 0, |x| = -x
$\text{f}(\text{x})=\frac{\text{x}}{1+|\text{x}|}$
$=\frac{\text{x}}{1+\text{x}}$
$\Rightarrow\text{f}'(\text{x})=\frac{(1-\text{x})1-\text{x}(-1)}{(1-\text{x})^2}$
$=\frac{1}{(1-\text{x})^2}>0,\forall\ \text{x}\in\text{R}$
So, f(x) is strictly increasing when x < 0.
Thus, f(x) is strictly increasing on R.

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