Question: The function $f(x)=\frac{4 x^{3}-3 x^{2}}{6}-2 \sin x+(2 x-1) \cos x:$
(1) increases in $\left[\frac{1}{2}, \infty\right)$
(2) decreases $\left(-\infty, \frac{1}{2}\right]$
(3) increases in $\left(-\infty, \frac{1}{2}\right]$
(4) decreases $\left[\frac{1}{2}, \infty\right)$
Correct Option: 1,
Solution:
$f^{\prime}(x)=(2 x-1)(x-\sin x)$
$\Rightarrow f^{\prime}(x) \geq 0$ in $x \in\left[\frac{1}{2}, \infty\right)$
