The sum of absolute maximum and absolute minimum values of the fu… - Vidyadip

MCQ

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

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

✓

Answer

Correct option: B.

$3+\frac{1}{2}(1+2 \cos (1)) \sin (1)$

b
$f(x)=\left|2 x^{2}+3 x-2\right|+\sin x \cos x$ $f(x)=|(2 x-1)(x+2)|+\sin x \cos x$

$f^{\prime}(x)=\left\{\begin{array}{cl}4 x+3+\frac{\cos 2 x}{4}, & \frac{1}{2} < x < 1 \\ -(4 x+3)+\frac{\cos 2 x}{4}, & 0 \leq x < \frac{1}{2}\end{array}\right.$

For $0 \leq x <\frac{1}{2} \Rightarrow f ^{\prime}( x )<0$

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

$f ( x )$ local minima at $x =\frac{1}{2}$ and

local maxima at $x=1$

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

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