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One of the root equation cosx-x+1/2=0 li...

One of the root equation `cosx-x+1/2=0` lies in the interval (a)`(0,pi/2)` (b) `(-pi/(2,0))` (c) `(pi/2,pi)` (d) `(pi,(3pi)/2)`

A

`(0,(pi)/(2))`

B

`((pi)/(2),pi)`

C

`(pi,(3pi)/(2))`

D

`((3pi)/(2),2pi)`

Text Solution

Verified by Experts

The correct Answer is:
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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