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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions3 Marks
Question
Without using the derivative, show that the function f(x) = |x| is
  1. Strictly increasing in $(0,\infty)$
  2. Strictly decreasing in $(-\infty,0)$
✓

Answer

We have,
$\text{f}(\text{x})=|\text{x}|=\begin{cases}\text{x},&\text{x}>0\\\text{-x},&\text{x}<0\end{cases}$
  1. Let $\text{x}_1,\text{x}_2\in(0,\infty)$ and $\text{x}_1>\text{x}_2$
$\Rightarrow\text{f}(\text{x}_1)>\text{f}(\text{x}_2)$

So, f(x) is increasing in $(0,\infty).$
  1. Let $\text{x}_1,\text{x}_2\in (-\infty,0)$ and $\text{x}_1>\text{x}_2$
$\Rightarrow-\text{x}_1<-\text{x}_2$

$\Rightarrow\text{f}(\text{x}_1)<\text{f}(\text{x}_2)$

So, f(x) is decreasing on $(-\infty,0).$

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