STD 12 - 6. Application of derivatives — JEE कक्षा 11 साइन्स — Question

$f^{\prime}(x)=\left(2 x^{2}-x\right)-2 \cos x+2 \cos x-\sin x(2 x-1)$

$\quad=(2 x-1)(x-\sin x)$

for $x>0, x-\sin x>0$

$\quad x<0, x-\sin x<0$

for $x \in(-\infty, 0] \cup\left[\frac{1}{2}, \infty\right), f^{\prime}(x) \geq 0$

for $x \in\left[0, \frac{1}{2}\right], f^{\prime}(x) \leq 0$

$\Rightarrow \quad f(x)$ increases in $\left[\frac{1}{2}, \infty\right)$