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  • Please solve RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 25 maths textbook solution

Please solve RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 25 maths textbook solution

Answers (1)

Answer:

f(x) \text { is increasing on } (-\frac{\pi }{3},\frac{\pi }{3})

Given:

f(x)=-\frac{x}{2}+sin\: x

To prove:

\text { We have to determine whether } f(x) \text { is increasing or decreasing on } (-\frac{\pi }{3},\frac{\pi }{3})

Hint:

For f(x) to be increasing we must have f ’(x) > 0 and f ‘(x) < 0 for decreasing.

Solution:

We have

f(x)=-\frac{x}{2}+sin\: x

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

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(-\frac{x}{2}+\sin x\right) \\ &\Rightarrow f^{\prime}(x)=-\frac{1}{2}+\cos x \\ &\text { Now, } x \in\left(-\frac{\pi}{3}, \frac{\pi}{3}\right) \\ &\Rightarrow-\frac{\pi}{3}<x<\frac{\pi}{3} \end{aligned}

\begin{aligned} &\Rightarrow \cos \left(-\frac{\pi}{3}\right)<\cos x<\cos \frac{\pi}{3} \\ &\Rightarrow \frac{1}{2}<\cos x,\left[\therefore \cos \left(-\frac{\pi}{3}\right)=\cos \frac{\pi}{3}=\frac{1}{2}\right] \\ &\Rightarrow-\frac{1}{2}+\cos x>0 \\ &\Rightarrow f^{\prime}(x)>0 \end{aligned}

\text { Hence } f(x) \text { is increasing function for x } \in \: (-\frac{\pi }{3},\frac{\pi }{3})

Posted by

Gurleen Kaur

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