The function f(x) = 4 x ^ 3 - 3 x ^ 2 6 - 2 x + (2x - 1) x :

f\left(x\right) = \dfrac{4{x}^{3} - 3{x}^{2}}{6} - 2\sin x + \left(2x - 1\right)\cos x$<p>${f}^{'}\left(x\right) = \left(2{x}^{2} - x\right) - 2\cos x + 2\cos x - \sin x\left(2x - 1\right)$.</p><p>$= \left(2x - 1\right)\left(x - \sin x\right)$</p><p>for$x > 0, x - \sin x > 0$</p><p>$x < 0, x - \sin x < 0$</p><p>for$x \in \left(- \infty , 0\right] \cup \left[\dfrac{1}{2}, \infty\right), {f}^{'}\left(x\right) \ge 0$</p><p>for$x \in \left[0, \dfrac{1}{2}\right], {f}^{'}\left(x\right) \le 0$</p><p>$⇒ f\left(x\right)$increases in \left[\dfrac{1}{2}, \infty\right)$