The fact that we're given our first derivative means we're going to set each of those terms equal to 0. So sine X = 1 when X is π halves cosine X equal negative 1/2 twice in a unit circle at 2π thirds and also at 4 Pi thirds. So taking into account the endpoints that are given to us, we have 5 locations, 0 Pi halves, 2π thirds, 4 Pi thirds, and 2π. If we test a value in between 0 and π halves, say π force, we plug it into the first derivative, we realize that's going to be a negative value in between π halves and 2π thirds. I chose 7 Pi twelfths to give it a try, and I grabbed my calculator and when I did 7 Pi twelfths I realized it was negative. You could also realize in between π halves and 2π thirds that your sine is going to be positive, but it's going to be less than one. So a positive number less than one is going to be negative. And then two, our cosine between π halves and 2π thirds is going to be fairly small. At 2π thirds, we know it's going to be negative 1/2, so it's got to be less than that. So two times something smaller than negative 1/2 + 1 would be a positive. So a negative times a positive would be a negative in between 2π thirds and four Pi thirds. If you use π you get sine of π is 0 - 1, so there's a -2 cosine π cosine π is -1 so you get a negative. So negative times negative makes that a positive within that interval. And then between 4 Pi thirds and 2π, use one of your other points and you're going to end up with a negative. So you could use four Pi. You could use three Pi halves. Three Pi halves would be an easy one. Sine of three Pi halves is -1, so a negative value. Cosine of three Pi halves is 0. So a negative times a positive is a negative. So now we know that our original function is decreasing from zero to π halves and it's also decreasing from Pi halves to 2π thirds. At 2π thirds it's going to increase and at 4 Pi thirds it's going to decrease. So on what interval is it increasing? It's increasing where our first derivatives positive, so 2π thirds to four Pi thirds. On what interval is it decreasing? It's decreasing from zero to π halves and it's decreasing from Pi halves to 2π thirds and it's decreasing from 4 Pi thirds to 2π. At what points, if any, do we have a local Max? We're going to have a local Max here because it's higher than any other point. We're also going to have a local Max at that four Pi thirds, so zero and four Pi thirds. When do we have a local min? Local min's going to occur at 2π thirds and also at 2π.