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Question
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
Solution
\[f\left( x \right) = x\left| x \right|, x \in R\]
\[\text { Case I: When x } \geq 0\]
\[f\left( x \right) = x\left| x \right| = x\left( x \right) = x^2 \]
\[ \Rightarrow f'\left( x \right) = 2x \geq 0 \forall x \geq 0\]
\[\text { So,} f\left( x \right)\text { is increasing for x } \geq 0 . \]
\[\text { Case II: When } x < 0\]
\[f\left( x \right) = x\left| x \right| = x\left( - x \right) = - x^2 \]
\[ \Rightarrow f'\left( x \right) = - 2x \geq 0 \forall x < 0\]
\[\text { So, }f\left( x \right)\text { is increasing for } x < 0 . \]
\[\text { Hence }, f\left( x \right)\text { is increasing for x } \in R . \]
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