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Differential Calculus

Blazing goIITian

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Post:	1251

2 Aug 2008 11:45:42 IST

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Easy - monotonicity

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$f(x) = \phi (\sin x ) + \phi(\cos x) \\ \\<br/>and \ \\ \\<br/>\phi$

Find the interval of monotonicity of f(x)

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Hari Shankar

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Joined: 28 Feb 2007

Posts: 2173

2 Aug 2008 13:05:17 IST

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$f(x) = \phi (\sin x) + \phi (\cos x) \Rightarrow f$

We are given that

$\phi$

Consider the interval $\left[0, \frac{\pi}{4} \right ]$ .

Now, $x \in \left[0, \frac{\pi}{4} \right ] \Rightarrow \frac{\pi}{2} - x \in \left[\frac{\pi}{4}, \frac{\pi}{2} \right ]$

Hence $\cos x \ge \sin x \ge 0$

Also, $\phi " (\sin x) < 0 \Rightarrow \sin x_1 > \sin x_2 \equiv \phi$

$\sin \left(\frac{\pi}{2} - x \right) > \sin x \Rightarrow \phi$

Thus, $\cos x \ \phi$

The opposite happens when $x \in \left [\frac{\pi}{4}, \frac{\pi}{2} \right ]$

Thus we have f is increasing in the first half and decreasing in the second quadrant

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