Show that function f(x)= x is increasing in (0, 2 ) - Vidyadip

Question

Show that function $f(x)=\tan x$ is increasing in $\left(0, \frac{\pi}{2}\right)$

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Answer

$f(x)=\tan x$
Differentlating w.r.t. $x$,
$
\begin{aligned}
& f^{\prime}(x)=\sec ^2 x=(\sec x)^2 \\
& \operatorname{In}\left(0, \frac{\pi}{2}\right) \cdot \sec ^2 x>0 \\
\therefore & f^{\prime}(x)>0 \text { for all } x \in\left(0, \frac{\pi}{2}\right) \\
\therefore & f(x)=\tan x \text { is increasing in }\left(0, \frac{\pi}{2}\right)
\end{aligned}
$

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