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Q.
The sum of absolute maximum and absolute minimum values of the function $f(x)=\left|2 x^{2}+3 x-2\right|+\sin x \cos x$ in the interval $[0,1]$ is :
$f(x)=\left|2 x^{2}+3 x-2\right|+\sin x \cos x$
$f(x)=|(2 x-1)(x+2)|+\sin x \cos x$
$f^{\prime}(x)=\begin{cases}4 x+3+\frac{\cos 2 x}{4}, & \frac{1}{2}< x< 1 \\ -(4 x+3)+\frac{\cos 2 x}{4}, & 0 \leq x< \frac{1}{2}\end{cases}$
For $0 \leq x< \frac{1}{2} \Rightarrow f^{\prime}(x)< 0$
For $\frac{1}{2}< x \leq 1 \Rightarrow f^{\prime}(x) >0$
$f ( x )$ local minima at $x =\frac{1}{2}$ and
local maxima at $x =1$
$f\left(\frac{1}{2}\right)+f(1)=3+\frac{1}{2}(1+2 \cos 1) \sin 1$