1.

The smallest positive root of the equation `tanx-x=0`lies in`(0,pi/2)`(b) `(pi/2,pi)``(pi,(3pi)/2)`(d) `((3pi)/2,2pi)`A. `(0,(pi)/(2))`B. `((pi)/(2),pi)`C. `(pi,(3pi)/(2))`D. `((3pi)/(2),2pi)`

Answer» Correct Answer - c
Let ` f (x) = tan x - x `
We know, for ` 0 lt x lt (pi)/(2)`
` rArr " " tanx gt x `
` therefore " " f (x) = tan x - x ` has no root in ` ( 0 , pi//2)`
For ` pi//2 lt x lt pi , tan x ` is negative.
` therefore " " f (x) = tan x - x lt 0 `
So, ` " " f(x) = 0 ` has no root in ` ((pi)/(2), pi )`
For ` " " ( 3pi )/(2) lt x lt 2pi, tan x ` is negative.
` therefore " " f (x) = tan x - x lt 0 `
So, ` f (x) = 0` has no root in ` (( 3pi)/(2), 2pi)`
We have, ` f(pi) = 0 -pi lt 0 `
and ` f((3pi)/(2)) = tan ""(3pi)/(2) - ( 3pi)/(2) gt 0 `
` therefore f (x) =0` has at least one root between `pi and ( 3pi )/(2)`


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