The sum of absolute maximum and absolute minimum values of the...

Given $f\left(x\right) = \left|2{x}^{2} + 3x - 2\right| + \sin x \cos x$

$⇒ f\left(x\right) = \left|\left(2x - 1\right)\left(x + 2\right)\right| + \sin x \cos x$

Now, {f}^{'}\left(x\right) = \left\{\begin{cases}4x + 3 + \dfrac{\cos 2x}{4}, & \dfrac{1}{2} < x < 1 \\ - \left(4x + 3\right) + \dfrac{\cos 2x}{4}, & 0 \le x < \dfrac{1}{2}\end{cases}\right)

For $0 \le x < \dfrac{1}{2} ⇒ {f}^{'}\left(x\right) < 0$

For $\dfrac{1}{2} 0$

So $f\left(x\right)$ has minima at $x = \dfrac{1}{2}$ and maxima at $x = 1$

Hence, $f\left(\dfrac{1}{2}\right) + f\left(1\right) = 3 + \dfrac{1}{2}\left(1 + 2\cos 1\right)\sin 1$