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  • Please solve RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 34 maths textbook solution

Please solve RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 34 maths textbook solution

Answers (1)

Answer:

f(x) \text { is an increasing function on } (0,\frac{\pi }{2})

Given:

f(x)=x^{2}-xsin\: x

To prove:

\text { We have to show that } f(x) \text { is an increasing function on } (0,\frac{\pi }{2})

Hint:

For f(x) to be increasing we must have f’(x) > 0.

Solution:

Given

f(x)=x^{2}-xsin\: x

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

\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(x^{2}-x \sin x\right)

\begin{aligned} &\Rightarrow f^{\prime}(x)=2 x-\sin x-x \cos x,\left[\therefore \frac{d}{d x} u v=v \frac{d u}{d x} \cdot u \frac{d v}{d x}\right] \\ &\text { Now, } x \in\left(0, \frac{\pi}{2}\right) \\ &\Rightarrow 0 \leq \sin x \leq 1 \\ &\Rightarrow 0 \leq \cos x \leq 1 \\ &\Rightarrow 2 x-\sin x-x \cos x>0 \\ &\Rightarrow f^{\prime}(x)>0 \end{aligned}

\text { Hence } f(x) \text { is an increasing function on } (0,\frac{\pi }{2})

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Gurleen Kaur

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