Question

The smallest positive root of the equation tan x – x = 0 lies in

• A. I quadrant
• B. II quadrant
• C. III quadrant
• D. IV quadrant

Answer 1

The correct option is C III quadrant

Let f(x) = tan x -x
When 0 < x < $\frac{\pi }{2}$, tan x > x $\Rightarrow$ f(x) > 0
$\therefore f\left(x\right)=0$ has no root in I quadrant
In the II and IV quadrant tan x is negative
$\therefore$ f(x) < 0 and f(x) = 0 has no solution in these quadrants
Next $f\left(\pi \right)=tan\pi -\pi$ <0
$f\left(x\right)\to \infty asx\to \frac{3\pi }{2}$
$\therefore f\left(x\right)=0$ has at least one root in the III quadrant.