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Show that 2sinx+tanx≥3x, where 0≤x< π/2 ...

Show that 2sinx+tanx≥3x, where 0≤x< π/2 ​

Text Solution

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Letf(x) =2sinx+tanx-3x
`therefore f(x) =2cosx +sec^(2)x-3`
`=sec^(2)x(2 cos^(3)x-3 cos^(2)-3)`
`=sec(2)x(2 cos^(3)x-3 cos^(2)x+1)`
`=sec^(2)x(1-cos)^(2)(1+cosx)`
`therefore f(x) ge 0, forall x in (0,pi//2)`
`therefore 2 sinx + tanx ge 3x` where `0 le x lt (pi)/(2)`
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