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# Prove that the following functions are not monotonic in the intervals given. $\tan x + \cot x$ on $[0 , \large\frac{\pi}{2}]$

Note: This is part 4th of a 4 part question, split as 4 separate questions here.

Toolbox:
• If $f(x)$ is defined on an open interval $I$ and $f'(x)$ takes both positive and negative values on $I$, then $f$ is not monotonic on $I$
$f(x)=\tan x +\cot x$ on $(0, \large\frac{\pi}{2})$
Step 1:
$f'(x)=\sec^2 x -cosec ^2 x$
$\qquad=\large\frac{1}{\cos ^2 x}-\frac{1}{\sin ^2 x}$
$\qquad=\large\frac{\sin ^2 x -\cos ^2 x}{\sin ^2 x \cos ^2 x}$
$\qquad=(\sin x + \cos x)(\sin x-\cos x)$
Step 2:
When $0 < x < \large\frac{\pi}{4} \; $$\sin x + \cos x >0 and \sin x -\cos x <0 \therefore\; f'(x) <0 When \large\frac{\pi}{4}$$ < x < \large\frac{\pi}{2} \;$$\sin x + \cos x >0$ and $f'(x) >0$
$f'(x)$ changes in sigh in $(0, \large\frac{\pi}{2})$
$\therefore$ it is not monotonic on $(0, \large\frac{\pi}{2})$

edited Jul 30, 2013 by meena.p