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  • please solve rd sharma class 12 chapter 16 Increasing and Decreasing Functions exercise Multiple choice question , question 8 maths textbook solution

please solve rd sharma class 12 chapter 16 Increasing and Decreasing Functions exercise Multiple choice question , question 8 maths textbook solution

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Correct option (a)

Hint: If g(x)  is increasing on \left ( 0,\frac{\pi }{2} \right ) thenx<y for all x,y\epsilon \left ( 0,\frac{\pi }{2} \right )

Given: f(x)=\tan^{-1}\left ( g\left ( x \right ) \right ) , where g\left ( x \right ) is monotonically increasing for 0<x<\frac{\pi}{2}

Explanation: We have,

g\left ( x \right ) is monotonically increasing on \left ( 0,\frac{\pi }{2} \right )

So, x<y for all x,y\epsilon \left ( 0,\frac{\pi }{2} \right )

\begin{aligned} &\Rightarrow g(x)<g(y) \\ &\Rightarrow \tan ^{-1} g(x)<\tan ^{-1} g(y) \end{aligned}

\begin{aligned} &\Rightarrow f(x)<f(y) \\ \end{aligned} for all x,y\epsilon \left ( 0,\frac{\pi }{2} \right )

Thus, f(x)  is increasing on\left ( 0,\frac{\pi }{2} \right )

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