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If equation 2x^(3)-6x+2sina+3=0,a in (0,...

If equation `2x^(3)-6x+2sina+3=0,a in (0,pi)` has only one real root, then the largest interval in which a lies is

A

`(0,(pi)/(6))`

B

`((pi)/(6),(pi)/(3))`

C

`((pi)/(6),(5pi)/(6))`

D

`((5pi)/(6),pi)`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `f(x)=2x^(3)-6x+2sin a+3`
`rArr" "f'(x)=6x^(2)-6=0 rArr x =pm 1`
Since equation f(x) = 0 has only one real root ltbtgt `f(-1),f(1)gt0`
`rArr" "(2sin a+7)(2sin a-1)gt0`
`rArr" "sina gt(1)/(2)rArr a in ((pi)/(6),(5pi)/(6))`
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