The function

Question:

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

  1. (1) increases in $\left[\frac{1}{2}, \infty\right)$

  2. (2) decreases $\left(-\infty, \frac{1}{2}\right]$

  3. (3) increases in $\left(-\infty, \frac{1}{2}\right]$

  4. (4) decreases $\left[\frac{1}{2}, \infty\right)$


Correct Option: 1,

Solution:

$f^{\prime}(x)=(2 x-1)(x-\sin x)$

$\Rightarrow f^{\prime}(x) \geq 0$ in $x \in\left[\frac{1}{2}, \infty\right)$

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