Question

The function
$f(x)=\frac{4x^3-3x^2}{6}-2\sin x+(2x-1)\cos x$

• A. increases in $\left[\frac{1}{2},\infty \right)$
• B. increases in $\left(-\infty ,\frac{1}{2}\right]$
• C. decreases in $\left[\frac{1}{2},\infty \right)$
• D. decreases in $\left(-\infty ,\frac{1}{2}\right]$

Answer 1

Answer: (A)

$f(x)=\frac{4x^3-3x^2}{6}-2\sin x+(2x-1)\cos x$
$f^{\prime }(x)=\left(2x^2-x\right)-2\cos x+2\cos x-\sin x(2x-1)$
$=(2x-1)(x-\sin x)$
for $x>0,x-\sin x>0$
$x<0,x-\sin x<0$
for $x\in (-\infty ,0]\cup \left[\frac{1}{2},\infty \right),f^{\prime }(x)\ge 0$
for $x\in \left[0,\frac{1}{2}\right],f^{\prime }(x)\le 0$
$\Rightarrow f(x)$ increases in $\left[\frac{1}{2},\infty \right)$