Mathematics - Application of Derivative Question with Solution | TestHub

$f\left(x\right)=\frac{4x^3-3x^2}{6}-2\sin x+\left(2x-1\right)\cos x$

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

$=\left(2x-1\right)\left(x-\sin x\right)$

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

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

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

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

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