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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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