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Please solve RD Sharma class 12 chapter 16 Increasing and Decreasing Function Excercise Fill in the blanks Question 8: Maths Textbook Solution.

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Answer: The value of k is \frac{1}{3} 

Hint: The function f(x)  is increasing if f'(x)>0 .

Given: f(x)=3\sin x-4\sin^{3}x

Explanation: It is given that

f(x)=3\sin x-4\sin ^{3}x=\sin 3x                          …(i)

Differentiate (i)  with respect to x

f'(x)=3\cos 3x

\because f(x)  is increasing.

\Rightarrow f'(x)>0

\Rightarrow 3\cos 3x>0

Now, if    3x\epsilon \left [ -\frac{\pi }{2}, \frac{\pi }{2}\right ]

\Rightarrow x\epsilon \left [ -\frac{\pi }{6}, \frac{\pi }{6}\right ]

Thus, the length of largest interval is   \frac{\pi }{6}-\left ( -\frac{\pi }{6} \right )=\frac{2\pi }{6}=\frac{\pi }{3}

\Rightarrow k=\frac{1}{3}

\because k\pi  is length of largest interval in which f(x)  is increasing)

Thus, the value of k is \frac{1}{3}

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