Find the intervals in which f(x) is increasing or decreasing: f(x… - Vidyadip

Question

Find the intervals in which f(x) is increasing or decreasing:
$\text{f}(\text{x})=\text{x}|\text{x}|,\text{x}\in\text{R}$

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Answer

$\text{f}(\text{x})=\text{x}|\text{x}|,\text{x}\in\text{R}$Case I:
When $\text{x}\geq0$ $\text{f}(\text{x})=\text{x}|\text{x}|=\text{x}(\text{x})=\text{x}^2$ $\Rightarrow\text{f}'(\text{x})=2\text{x}\geq0\ \forall\ \text{x}\geq0$ So, f(x) is increasing for $\text{x}\geq0.$Case II:
When $\text{x}<0$ $\text{f}(\text{x})=\text{x}|\text{x}|=\text{x}(-\text{x})=-\text{x}^2$ $\Rightarrow\text{f}'(\text{x})=-2\text{x}\geq0\ \forall\ \text{x}<0$ So, f(x) is increasing for x < 0. Hence f(x) is increasing for $\text{x}\in\text{R}.$

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