# Determine if a function is increasing/decreasing at a particular point

I'm getting better at using trig functions, but this problem has me up against a wall.

As I understand it, to find if a function is increasing or decreasing over a given interval, select a number inside that interval and if the result is greater or less than zero, the interval is increasing and decreasing, respectively.

Given $f(x)=5x+10 \sin x$ on the interval $(0, 2\pi)$, find the open intervals where the function is increasing or decreasing.

The given intervals are $(0, \frac{2\pi}{3}), (\frac{2\pi}{3}, \frac{4\pi}{3}), (\frac{4\pi}{3}, 2\pi).$

On the first interval, I selected $\frac{\pi}{2}$ as the test for x. Thus $$f(x)= 5x+10 \sin x \Rightarrow 5(\frac{\pi}{2}) + 10 \sin(\frac{\pi}{2})\Rightarrow \frac{5\pi}{2}+10$$

$\frac{5\pi}{2}+10 > 0$, so the function should be increasing at that point

The second interval test variable was $\pi$, and thus $5(\pi)+10 \sin(\pi)= 5\pi > 0$ So the interval of $(\frac{2\pi}{3}, \frac{4\pi}{3})$ should be increasing.

The third interval test variable was $\frac{3\pi}{2}$, and $5(\frac{3\pi}{2})+10 \sin(\frac{3\pi}{2})= \frac{15\pi}{2}-10 > 0$, so that interval is increasing as well.

However, the WebAssign site didn't accept those answers. What is missing with my logic and work?

-