Mathematics - Application of Derivative Question with Solution | TestHub

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

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

Now, $f^{\text{'}}\left(x\right)=\left\{\begin{array}{cc}4x+3+\frac{\cos 2x}{4},& \frac{1}{2}<x<1\\ -\left(4x+3\right)+\frac{\cos 2x}{4},& 0\le x<\frac{1}{2}\end{array}\right.$

For $0\le x<\frac{1}{2}\Rightarrow f^{\text{'}}\left(x\right)<0$

For $\frac{1}{2}<x\le 1\Rightarrow f^{\text{'}}\left(x\right)>0$

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

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