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Solution for Show that the Function F(X) = Cot − L(Sinx + Cosx) is Decreasing on ( 0 , π 4 ) and Increasing on ( 0 , π 4 ) ? - CBSE (Commerce) Class 12 - Mathematics

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Question

Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?

Solution

\[\text { We have,} \]

\[f\left( x \right) = \cot^{- 1} \left( \sin x + \cos x \right)\]

\[ \Rightarrow f'\left( x \right) = \frac{- 1}{1 + \left( \sin x + \cos x \right)^2} \times \left( \cos x - \sin x \right)\]

\[ = \frac{\sin x - \cos x}{1 + \sin^2 x + \cos^2 x + 2\sin x\cos x}\]

\[ = \frac{\sin x  - \cos x}{1 + 1 + 2\sin x\cos x}\]

\[ = \frac{\sin x - \cos x}{2 + 2\sin x\cos x}\]

\[ = \frac{1}{2} \times \frac{\sin x - \cos x}{1 + \sin x\cos x}\]

\[\text { For } f\left( x \right) \text { to be decreasing, we must have }\]

\[f'\left( x \right) < 0\]

\[ \Rightarrow \frac{1}{2} \times \frac{\sin x - \cos x}{1 + \sin x\cos x} < 0\]

\[ \Rightarrow \frac{\sin x - \cos x }{1 + \sin x\cos x} < 0\]

\[ \Rightarrow \sin x - \cos x < 0 \left( \text { In first quadrant } \right)\]

\[ \Rightarrow \sin x < \cos x\]

\[ \Rightarrow \tan x < 1\]

\[ \Rightarrow 0 < x < \frac{\pi}{4}\]

\[So, f\left( x \right) \text { is decreasing on } \left( 0, \frac{\pi}{4} \right) . \]

\[\text { For } f\left( x \right) \text { to be increasing, we must have } \]

\[f'\left( x \right) > 0\]

\[ \Rightarrow \frac{1}{2} \times \frac{\sin x - \cos x}{1 + \sin x\cos x} > 0\]

\[ \Rightarrow \frac{\sin x - \cos x}{1 + \sin x\cos x} > 0\]

\[ \Rightarrow \sin x - \cos x > 0 \left(\text {  In first quadrant } \right)\]

\[ \Rightarrow \sin x > \cos x\]

\[ \Rightarrow \tan x > 1\]

\[ \Rightarrow \frac{\pi}{4} < x < \frac{\pi}{2}\]

\[\text { So,} f\left( x \right) \text { is increasing on } \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]

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Solution Show that the Function F(X) = Cot − L(Sinx + Cosx) is Decreasing on ( 0 , π 4 ) and Increasing on ( 0 , π 4 ) ? Concept: Increasing and Decreasing Functions.
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