Critical Points question - How do I write down this answer?

[WatkinsonMolly]

This question is from my highschool calculus homework, and I've got a lead to the answer, but I don't really know how to really best write it. I have y' calculated to = 1 + 2sin(2x), but for starters, this has an infinite amount of zeroes and the book doesn't give an interval to work within, so how do I really notate the critical points since there are an infinite amount on the domain? Also, the zeroes all seem to be at funky intervals; for instance, the one closest to 2 is = 1.8325957 so I was wondering if I missed a step somewhere to help me get the zeroes in terms of pi. Thanks in advance for the help!

 

[galactus]

\(\displaystyle \frac{7\pi}{12}\approx 1.832....\)

You could write the solutions in terms of multiples of Pi.

\(\displaystyle x=C\pi +\frac{7\pi}{12}, \;\ x=C\pi -\frac{\pi}{12}\)


\(\displaystyle 1+2sin(2x)=0\)

\(\displaystyle x=\frac{sin^{-1}(-1/2)}{2}=\frac{-\pi}{12}\)

 

[WatkinsonMolly]

Thank you so much! I tried to figure out what ratio of pi the answer was, but I couldn't figure it out. I appreciate that so much!