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Provide solution for RD Sharma maths class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 39 subquestion (i)

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 f(x)=x |x| is an increasing function for all real values.


  f(x)=x |x|

To prove:

 We have to find the interval in which f(x) is an increasing or decreasing.


 For f(x) to be increasing function we must have f’(x) > 0.


Here we have

f(x)=f(x)=x|x|, x \in R

We know that

f(x)=\left\{-x^{2}, \quad \text { if } x<0 \& x^{2}, \quad \text { if } x>0\right.

On differentiating f(x) w.r.t x we get

\begin{aligned} &f^{\prime}(x)=\{-2 x, \quad \text { if } x<0 \& 2 x, \quad \text { if } x>0 \\ &\Rightarrow f^{\prime}(x)>0 \text { for all value of } x . \end{aligned}

Hence f(x) is increasing function for all value of real values.

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Gurleen Kaur

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