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Show that 2 sin x + tan x >= 3 x where 0...

Show that `2 sin x + tan x >= 3 x` where `0 < x < pi/2.`

Text Solution

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Let ` " " y = f(x_) = 2 sin x + 2 tan x - 3 x `
` rArr " " f' (x) = 2 cos x + 2 sec ^(2) x - 3 `
For `" " 0 le x lt pi//2, f' (x) gt 0`
Thus, ` f (x)` is increasing
When `" "x ge 0, f (x) ge f (0)`
`rArr " " 2 sin x + 2 tan x - 3 x ge 0 + 0- 0`
` rArr " " 2 sin x + 2 tan x ge 3x`
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