find the intervals on which f is increasing and the intervals on which it is decreasing. f(x)= - 2 cos(x)-x…

find the intervals on which f is increasing and the intervals on which it is decreasing. f(x)= - 2 cos(x)-x on 0,π select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the function is increasing on the open interval(s) and decreasing on the open interval(s) (simplify your answers. use a comma to separate answers as needed. type your answers in interval notation. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) b. the function is increasing on the open interval(s). the function is never decreasing. (simplify your answers. use a comma to separate answers as needed. type your answers in interval notation. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) c. the function is decreasing on the open interval(s). the function is never increasing. (simplify your answer. use a comma to separate answers as needed. type your answers in interval notation. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) d. the function is never increasing or decreasing.

find the intervals on which f is increasing and the intervals on which it is decreasing. f(x)= - 2 cos(x)-x on 0,π select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the function is increasing on the open interval(s) and decreasing on the open interval(s) (simplify your answers. use a comma to separate answers as needed. type your answers in interval notation. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) b. the function is increasing on the open interval(s). the function is never decreasing. (simplify your answers. use a comma to separate answers as needed. type your answers in interval notation. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) c. the function is decreasing on the open interval(s). the function is never increasing. (simplify your answer. use a comma to separate answers as needed. type your answers in interval notation. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) d. the function is never increasing or decreasing.

Answer

Explanation:

Step1: Find the derivative of f(x)

$f'(x)=2\sin(x)- 1$

Step2: Set the derivative equal to zero

$2\sin(x)-1 = 0$ $\sin(x)=\frac{1}{2}$ On the interval $[0,\pi]$, $x = \frac{\pi}{6}$ and $x=\frac{5\pi}{6}$

Step3: Test intervals

  • For the interval $(0,\frac{\pi}{6})$, let's choose a test - point, say $x=\frac{\pi}{12}$. Then $f'(\frac{\pi}{12})=2\sin(\frac{\pi}{12})-1<0$. So $f(x)$ is decreasing on $(0,\frac{\pi}{6})$.
  • For the interval $(\frac{\pi}{6},\frac{5\pi}{6})$, let's choose $x = \frac{\pi}{2}$. Then $f'(\frac{\pi}{2})=2\sin(\frac{\pi}{2})-1 = 1>0$. So $f(x)$ is increasing on $(\frac{\pi}{6},\frac{5\pi}{6})$.
  • For the interval $(\frac{5\pi}{6},\pi)$, let's choose $x=\frac{11\pi}{12}$. Then $f'(\frac{11\pi}{12})=2\sin(\frac{11\pi}{12})-1<0$. So $f(x)$ is decreasing on $(\frac{5\pi}{6},\pi)$.

Answer:

A. The function is increasing on the open interval(s) $(\frac{\pi}{6},\frac{5\pi}{6})$ and decreasing on the open interval(s) $(0,\frac{\pi}{6}),(\frac{5\pi}{6},\pi)$