Question

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

• A. $3+\frac{\sin (1){\cos }^2(1/2)}{2}$
• B. $3+\frac{1}{2}(1+2\cos (1))\sin (1)$
• C. $5+\frac{1}{2}(\sin (1)+\sin (2))$
• D. $2+\sin \left(\frac{1}{2}\right)\cos \left(\frac{1}{2}\right)$

Answer 1

Answer: (B)

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