Question

The smallest positive root of the equation $\tan x-x=0$ lies in

• A. $(0,\pi /2)$
• B. $(\pi /2,\pi )$
• C. $(\pi ,\frac{3\pi }{2})$
• D. $(\frac{3\pi }{2},2\pi )$

Answer 1

The correct option is C $(\pi ,\frac{3\pi }{2})$

Drawing the graph for
$f\left(x\right)=x$ and
$g\left(x\right)=\tan x$


The slope of both the curves,
$g^{\prime }\left(x\right)={\sec }^{2}x{f}^{\prime }\left(x\right)=1$
As,
${\sec }^{2}x\ge 1\forall x\in (0,\pi /2)$
So,
$g\left(x\right)>f\left(x\right)\forall x\in (0,\pi /2)$

Hence the nearest points of intersection (roots of the equation) of both the graphs will be in $(\pi ,\frac{3\pi }{2})$.