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  • Need solution for RD Sharma maths class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 11

Need solution for RD Sharma maths class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 11

Answers (1)

Answer:

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

Given:

f(x) =cos^{2}\: x

To prove:

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

Hint:

Condition for f(x) to be decreasing that f’(x)<0.

Solution:

Given

f(x) =cos^{2}\: x

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

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(\cos ^{2} x\right) \\ &=2 \cos x(-\sin x) \\ &\Rightarrow f^{\prime}(x)=-2 \sin x \cos x \\ &\Rightarrow f^{\prime}(x)=-\sin 2 x \end{aligned}

Now, as given

x\: \in \: (0,\frac{\pi }{2})

\begin{aligned} &\Rightarrow 2 x \in(0, \pi) \\ &\Rightarrow \sin (2 x)>0 \\ &\Rightarrow-\sin (2 x)<0 \end{aligned}

By applying negative sign, change in comparison sign.

\begin{aligned} &\Rightarrow f'(x)< 0 \end{aligned}

Hence condition of f(x) to be decreasing.

Thus f(x) is decreasing on interval

x\: \in \: (0,\frac{\pi }{2}).

Posted by

Gurleen Kaur

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