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please solve rd sharma class 12 chapter 16 Increasing and Decreasing Functions exercise 16.1 , question 8 sub question b maths textbook solution

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f(x)is strictly increasing in (-\infty,0)


Use f(x)=|x|=\left\{\begin{array}{l} x, \text { if } x>0 \\ -x, \text { if } x<0 \end{array}\right.



To prove:

We have to prove that function f(x)=|x| is strictly increasing in (-\infty,0).


Let x_{1},x_{2}\in (-\infty,0 ) and x_{1}>x_{2}

Sincex_{1},x_{2}<0, therefore the function of x_{1},x_{2} is

           \begin{aligned} &-x_{1}<-x_{2} \\ &f\left(x_{1}\right)<f\left(x_{2}\right) \end{aligned}

Thus, x_{1}>x_{2} \Rightarrow f\left(x_{1}\right)<f\left(x_{2}\right) for all x_{1},x_{2}\in (-\infty,0 ).

So, f(x)is strictly increasing in (-\infty,0)

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