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Provide solution for RD Sharma maths class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 6

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f(x) is decreasing function or all x>0


f(x)=log_{a}x,\: 0< a< 1

To show:

We have to show that f(x) is decreasing function or all x>0.


\text { If } f^{\prime}(x)<0 \text { for all } x \in(a, b) \text { then } f(x) \text { is decreasing so first find } f^{\prime}(x) .



f(x)=log_{a}x,\: 0< a< 1

On differentiating both sides w.r.t x we get

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(\log _{a} x\right) \\ &=\frac{1}{x \log a} \\ &\text { As given } 0<a<1 \\ &\Rightarrow \log (a)<0 \text { and as } x>0 \\ &\Rightarrow \frac{1}{x}<0 \\ &\therefore f^{\prime(x)}=\frac{1}{x \log a}<0 \\ &\Rightarrow f^{\prime}(x)<0 \end{aligned}

Hence f(x) is decreasing for all x>0.

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Gurleen Kaur

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