The sum of absolute maximum and absolute minimum values of the function $\mathrm{f}\left(\mathrm{x}\right)=\left|2{\mathrm{x}}^2+3\mathrm{x}-2\right|+\sin \mathrm{xcos}\mathrm{x}$  in the interval [0, 1] is :

$\begin{array}{l}\mathrm{f}\left(\mathrm{x}\right)=\left|2{\mathrm{x}}^2+3\mathrm{x}-2\right|+\sin \mathrm{xcos}\mathrm{x}\\ \mathrm{f}\left(\mathrm{x}\right)=\left|\right(2\mathrm{x}-1\left)\right(\mathrm{x}+2\left)\right|+\sin \mathrm{xcos}\mathrm{x}\end{array}$
${\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=\left\{\begin{array}{cc}4\mathrm{x}+3+\frac{\cos 2\mathrm{x}}{4},& \frac{1}{2}<\mathrm{x}<1\\ -(4\mathrm{x}+3)+\frac{\cos 2\mathrm{x}}{4},& 0\le \mathrm{x}<\frac{1}{2}\end{array}\right.$
For $0\le \mathrm{x}<\frac{1}{2}\Rightarrow {\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)<0$
For $\frac{1}{2}<\mathrm{x}\le 1\Rightarrow {\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)>0$
f (x) local minima at $\mathrm{x}=\frac{1}{2}$ and 
local maxima at x = 1 
$\mathrm{f}\left(\frac{1}{2}\right)+\mathrm{f}\left(1\right)=3+\frac{1}{2}(1+2\cos 1)\sin 1$