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Prove that for x in [0, (pi)/(2)], sin x...

Prove that for `x in [0, (pi)/(2)], sin x + 2x ge (3x(x + 1))/(pi)`.

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Let `f (x) = sin x + 2x - ( 3x (x + 1))/( pi )`
On differentiating w.r.t. x, we get
`rArr f ' (x) = cos x + 2 - (( 6x + 3))/( pi )`
`rArr f '' (x) = - sin x - ( 6)/(pi) lt 0, AA x in [0, (pi)/(2)]`
`therefore, f'(x)` is decreasing for all `x in [0, (pi)/(2)]`.
`rArr " " f ' (x) gt 0" " [because x lt pi//2]`
` therefore f(x)` is increasing.
Thus, when ` x ge 0 , f (x) ge f(0)`
`rArr sin x + 2 x - ( 3x (x + 1))/(pi) ge -0`
`rArr sin x + 2x ge (3x ( x + 1))/(pi)`
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