Question

If $f\left(x\right)=\tan x,x\in [0,\frac{\pi }{5}]$ then show $\frac{\pi }{5}<f\left(\frac{\pi }{5}\right)<\frac{2\pi }{5}$.

Answer 1

We know that,

When $x>0$

$\tan x>x$

$\Rightarrow \tan \left(\frac{\pi }{5}\right)>\frac{\pi }{5}$

For upto x slightly greater than $1$ we can see that $2x>\tan x$

And,

as $\frac{\pi }{5}<1$

$\Rightarrow \frac{2\pi }{5}>\tan \left(\frac{\pi }{5}\right)$

$\Rightarrow \frac{\pi }{5}<\tan \left(\frac{\pi }{5}\right)<\frac{2\pi }{5}$.