Question

Find the x-coordinates where f '(x) = 0 for f(x) = 2x + sin(2x) in the interval [0, 2π].

so far I found f'(x)=2cos(2x)+2
cos(2x)=-1

Answer 1

so far good
do you know how to solve:
\[\cos(\theta)=-1 \text{ for } \theta \]

Answer 2

no:(

Answer 3

have you ever seen the unit circle before?

Answer 4

oh yes

Answer 5

Do x coordinates of the pairs represents cos
these are the numbers we want to look at
can you find when the x-coordinates on the unit circle will be -1?

Answer 6

The x coordinates * (not do)

Answer 7

so pie?

Answer 8

pi is going to be one solution

there is another solution
we were solving cos(2x)=-1 on [0,2pi]
but I replaced 2x with theta
so we had 0<=x<=2pi
and x is theta/2
so 0<=theta/2<=2pi
multiply 2 on both sides we have
0<=theta<=4pi
so we actually want to solve cos(theta)=-1 on [0,4pi]
but that isn't too terrible
we know pi is one solution in that interval
but pi+2pi is another


we wanted to solve for x not theta
but we know the relationship between x and theta is given by 2x=theta
so we have
\[2x=\pi \text{ or also } 2x=\pi+2\pi\]
simplify and solve for x

Answer 9

so that would be 2x=3pi
so x=pi/2 and x=3pi/2

Answer 10

sounds great to me :)

Answer 11

thanks again

Answer 12

np