BITSAT 2015 Paper

Let us assume the functions f(x) and g(x) given by f(x) = tanx – x and g(x) = x – sinx, for 0 < x < $\frac{\pi}{2}$ Now , f ' (x) = $\sec^2 x$ - 1 and g ' (x) = 1 - cos x r f ' (x) > 0 and g ' (x) > 0 for all x ∊ $\left(0,\frac{\pi}{2}\right)$ ⇒ f (x) > f (0) and g (x) > g (0) for all x ∊ $\left(0,\frac{\pi}{2}\right)$ ⇒ tan x - x > 0 and x - sin x > 0 for all x ∊ $\left(0,\frac{\pi}{2}\right)$ ⇒ tan x > x and x > sin x for all x ∊ $\left(0,\frac{\pi}{2}\right)$ ⇒ sin x < x < tan x for all x ∊ $\left(0,\frac{\pi}{2}\right)$